Permute Matrix Rows, Rearrange the Inverse Columns

1. Formula
   Observation

   The screen displays the $3\times 3$ matrix [a b c; d e f; g h i]

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

A

$A 3\times 3$ square matrix with elements a,b,c,d,e,f,g,h,i, whose inverse is known

$3\times 3$ square matrix

1. Formula
   Observation

   The screen displays the $3\times 3$ matrix [g h i; a b c; d e f]

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

N

Editorial N uses rows3,1,2 of A to form a new3×3 matrix.

$3\times 3$ square matrix

1. Formula
   Observation

   The screen displays [a' b' c'; d' e' f'; g' h' i'], and the audio explains that this is the inverse of A

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

$A^{-1}$

The inverse matrix of A, with elements denoted by primed variables a',b',c',d',e',f',g',h',i'

$3\times 3$ square matrix

1. Audio
   Observation

   The narration uses the original rows and corresponding inverse columns to identify the inverse after rearrangement.

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

I

Identity matrix, the result of multiplying a matrix by its inverse in the definition

$3\times 3$ square matrix

1. Formula
   Observation

   The screen shows the $3\times 3$ matrix $\begin{bmatrix} a&b&c\\ d&e&f\\ g&h&i \end{bmatrix}$, referred to as the "known matrix".

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

A

The original matrix with elements $a,b,c,d,e,f,g,h,i$.

$3\times 3$ matrix

1. Formula
   Observation

   The screen shows $\begin{bmatrix} a'&b'&c'\\ d'&e'&f'\\ g'&h'&i' \end{bmatrix}$ and states it is the inverse of $A$.

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

K

The inverse matrix of $A$, with elements $a',b',c',d',e',f',g',h',i'$.

$3\times 3$ matrix

1. Formula
   Observation

   The right side of the screen displays $\begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&1 \end{bmatrix}$.

2. Audio
   Observation

   The narration uses the original rows and corresponding inverse columns to identify the inverse after rearrangement.

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

$I_3$

The 3rd-order identity matrix.

$3\times 3$ matrix

1. Formula
   Observation

   The first row of the original matrix is $[a\ b\ c]$, marked in green on the screen.

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

[a\ b\ c]

The first row of matrix $A$.

Matrix row vector

1. Formula
   Observation

   The second row of the original matrix is $[d\ e\ f]$, marked in pink on the screen.

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

[d\ e\ f]

The second row of matrix $A$.

Matrix row vector

1. Formula
   Observation

   The third row of the original matrix is $[g\ h\ i]$, marked in cyan on the screen.

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

[g\ h\ i]

The third row of matrix $A$.

Matrix row vector

1. Formula
   Observation

   The first column of the inverse matrix is $\begin{bmatrix} a'\\ d'\\ g' \end{bmatrix}$.

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

$\begin{bmatrix} a'\\ d'\\ g' \end{bmatrix}$

The first column of matrix $K=A^{-1}$.

Matrix column vector

1. Formula
   Observation

   The second column of the inverse matrix is $\begin{bmatrix} b'\\ e'\\ h' \end{bmatrix}$.

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

$\begin{bmatrix} b'\\ e'\\ h' \end{bmatrix}$

The second column of matrix $K=A^{-1}$.

Matrix column vector

1. Audio
   Observation

   The narration uses the original rows and corresponding inverse columns to identify the inverse after rearrangement.

2. Formula
   Observation

   The source gives A and its inverse K and uses the identity criterion; editorial notation is AK=KA=I₃ with explicit row and column orientation.

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

The given $K=A$⁻¹ satisfies AK=KA=I₃. The source uses row-column pairing for the new inverse; editorial names avoid conflicting segment labels.

1. A is an invertible square matrix

1. Audio
   Observation

   The narration uses the original rows and corresponding inverse columns to identify the inverse after rearrangement.

2. Diagram
   Observation

   In the video, the speaker circles corresponding rows of A and N in green

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

The rows of N are rows3,1,2 of A. Editorial notation is N=PA with a permutation matrix P. A three-cycle can be decomposed into two swaps and is not a single row-swap elementary matrix.

1. Definition of Inverse Matrix

1. Formula
   Observation

   The screen states "The inverse of the known matrix $\begin{bmatrix} a&b&c\\ d&e&f\\ g&h&i \end{bmatrix}$ is $\begin{bmatrix} a'&b'&c'\\ d'&e'&f'\\ g'&h'&i' \end{bmatrix}$", and the right side shows $=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$.

2. Audio
   Observation

   The narration uses the original rows and corresponding inverse columns to identify the inverse after rearrangement.

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

The given $K=A$⁻¹ satisfies AK=KA=I₃. The source uses row-column pairing for the new inverse; editorial names avoid conflicting segment labels.

1. A and K are both3×3 matrices with $K=A$⁻¹

2. Editorial notation uses AK=I₃ for the given inverse relationship.

1. Audio
   Observation

   The narration uses the original rows and corresponding inverse columns to identify the inverse after rearrangement.

2. Diagram
   Observation

   The screen uses different colors to circle the rows of the original matrix and columns of the inverse matrix, showing a one-to-one correspondence between rows and columns.

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

At position(r,c), pair row r of N with column c of candidate C and compare with the corresponding identity entry.

1. Applicable to element-wise verification of matrix multiplication

2. In this problem, $r,c\in\{1,2,3\}$

1. Product of Inverse and Original Matrix Yields Identity Matrix

1. Audio
   Observation

   The narration uses the original rows and corresponding inverse columns to identify the inverse after rearrangement.

2. Diagram
   Observation

   In the screen, the column order of option (5) changes relative to the original inverse matrix, while the row order of the original matrix $A$ is also rearranged.

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

The video emphasizes a shortcut for solving: when the rows of the left matrix are swapped, the columns of the right matrix must be correspondingly swapped to maintain the identity matrix result after "row-column pairing". This problem utilizes this correspondence to quickly lock in the answer.

1. For N=PA, C=KP⁻¹; P is a row permutation matrix and $K=A$⁻¹.

1. "Row-Column Pairing" Rule for Matrix Multiplication

1. Audio
   Observation

   The narration uses the original rows and corresponding inverse columns to identify the inverse after rearrangement.

2. Diagram
   Observation

   The screen shows the rows of A and N circled to indicate correspondence, and writes out the structure of $A^{-1}$$A = I$

1. P and the full product notation are editorial. The model prematurely read option3; the actual final source selects5.

1. Expression

   $$N=\begin{bmatrix}g&h&i\\a&b&c\\d&e&f\end{bmatrix},\quad A=\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}$$
   Explanation

   Compare the given row ordering.

   Justification

   Editorial permutation verification from the given A,K and actual final option5; the full P formulas are not attributed to the source.

   Supplementary explanation
2. Expression

   $$N=PA,\quad P=\begin{bmatrix}0&0&1\\1&0&0\\0&1&0\end{bmatrix}$$
   Explanation

   P reorders rows3,1,2.

   Justification

   Editorial permutation verification from the given A,K and actual final option5; the full P formulas are not attributed to the source.

   Supplementary explanation
3. Expression

   $$N^{-1}=A^{-1}P^{-1}=KP^{-1}$$
   Explanation

   Reverse the invertible factors; retain the right-side permutation.

   Justification

   Editorial permutation verification from the given A,K and actual final option5; the full P formulas are not attributed to the source.

   Supplementary explanation
4. Expression

   $$P^{-1}=P^T=\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}$$
   Explanation

   Direct multiplication gives both identity products for P and its transpose.

   Justification

   Editorial permutation verification from the given A,K and actual final option5; the full P formulas are not attributed to the source.

   Supplementary explanation
5. Expression

   $$N^{-1}=K\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}$$
   Explanation

   Right multiplication places the old columns3,1,2 in the new result.

   Justification

   Editorial permutation verification from the given A,K and actual final option5; the full P formulas are not attributed to the source.

   Supplementary explanation
6. Expression

   $$N^{-1}=\begin{bmatrix}c'&a'&b'\\f'&d'&e'\\i'&g'&h'\end{bmatrix}$$
   Explanation

   The actual complete source option is5.

   Justification

   Editorial permutation verification from the given A,K and actual final option5; the full P formulas are not attributed to the source.

   Supplementary explanation

The new inverse uses columns3,1,2 of K, giving option5.

1. Audio
   Observation

   The narration uses the original rows and corresponding inverse columns to identify the inverse after rearrangement.

2. Diagram
   Observation

   Editorial positional reading pairs the second row of N, [a b c], with the second column [a′ d′ g′]ᵀ of actual option5.

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

1. Expression

   $$A=[a\ b\ c;\ d\ e\ f;\ g\ h\ i]\quad\Rightarrow\quad N=[g\ h\ i;\ a\ b\ c;\ d\ e\ f]$$
   Explanation

   The left matrix in the problem has undergone row rearrangement relative to the original $A$; the original first row $[a\ b\ c]$ is now the second row.

   Justification

   The screen directly shows the difference in row order between the problem matrix and the original matrix.

   Supplementary explanation
2. Expression

   $$(NC)_{22}=[a\ b\ c]\begin{bmatrix}a'\\d'\\g'\end{bmatrix}=1$$
   Explanation

   Row two of N pairs with column two of C, whose complete entries are a′,d′,g′.

   Justification

   The first diagonal pairing in AK moves to the second diagonal position.

   Supplementary explanation
3. Expression

   $$[a\ b\ c]\cdot\begin{bmatrix} a'\\ d'\\ g' \end{bmatrix}=1$$
   Explanation

   Originally, $[a\ b\ c]$ in $AK=I_3$ pairs with $\begin{bmatrix} a'\\ d'\\ g' \end{bmatrix}$; after the row moves to position2, this column also moves to position2.

   Justification

   Following the row-column pairing relationship of the original inverse matrix and applying the rule stated in the video that row swaps correspond to column swaps.

   Supplementary explanation
4. Expression

   $$C=\begin{bmatrix}c'&a'&b'\\f'&d'&e'\\i'&g'&h'\end{bmatrix}$$
   Explanation

   The middle column of option (5) is exactly $\begin{bmatrix} a'\\ d'\\ g' \end{bmatrix}$, which meets the above requirement.

   Justification

   All nine actual final option5 entries agree with the editorial KP⁻¹ computation.

   Supplementary explanation

Therefore, the answer to this problem should be (5).

1. Audio
   Observation

   The narration uses the original rows and corresponding inverse columns to identify the inverse after rearrangement.

2. Diagram
   Observation

   Editorial matching uses the actual N rows g,h,i; a,b,c; d,e,f and option5 columns c′f′i′; a′d′g′; b′e′h′.

1. Model transcription confused row positions; editorial correction uses the actual problem and final matrix, without alleging an author spoken error.

1. Expression

   $$[d\ e\ f]\cdot\begin{bmatrix}b'\\e'\\h'\end{bmatrix}=1$$
   Explanation

   Row3 of N pairs with column3 of C, inheriting the corresponding diagonal entry of AK.

   Justification

   AK=I₃; the reordering preserves these diagonal pairings and makes the other pairings zero.

   Supplementary explanation
2. Expression

   $$[g\ h\ i]\cdot\begin{bmatrix}c'\\f'\\i'\end{bmatrix}=1$$
   Explanation

   Row1 of N pairs with column1 of C, inheriting the corresponding diagonal entry of AK.

   Justification

   AK=I₃; the reordering preserves these diagonal pairings and makes the other pairings zero.

   Supplementary explanation
3. Expression

   $$[a\ b\ c]\cdot\begin{bmatrix}a'\\d'\\g'\end{bmatrix}=1$$
   Explanation

   Row2 of N pairs with column2 of C, inheriting the corresponding diagonal entry of AK.

   Justification

   AK=I₃; the reordering preserves these diagonal pairings and makes the other pairings zero.

   Supplementary explanation

All three sets of row-column pairings are consistent with the requirements of the identity matrix, further supporting that option (5) is the correct inverse matrix.

1. Formula
   Observation

   The screen displays the complete problem: Given the inverse of A is $A^{-1}$, find the inverse of N, listing five options

2. Audio
   Observation

   The narration uses the original rows and corresponding inverse columns to identify the inverse after rearrangement.

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

Given square matrix A = [a b c; d e f; g h i] and its inverse $A^{-1}$ = [a' b' c'; d' e' f'; g' h' i']. Which of the following options is the inverse of N = [g h i; a b c; d e f]?

1. $A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$

2. $A^{-1}$ = $\begin{bmatrix} a' & b' & c' \\ d' & e' & f' \\ g' & h' & i' \end{bmatrix}$

3. $N = \begin{bmatrix} g & h & i \\ a & b & c \\ d & e & f \end{bmatrix}$

Find the inverse matrix $N^{-1}$

1. Expression

   $$N=PA$$
   Explanation

   The first row of N is the third row of A, the second row is the first row of A, and the third row is the second row of A.

   Justification

   Check factor sides and the full identity relationship from the source row ordering and actual option5.

   Supplementary explanation
2. Expression

   $$N = PA$$
   Explanation

   Represent the row permutation as left-multiplication by an permutation matrix P.

   Justification

   Check factor sides and the full identity relationship from the source row ordering and actual option5.

   Supplementary explanation
3. Expression

   $$N^{-1} = A^{-1}P^{-1}$$
   Explanation

   Expand using the multiplicative property of inverse matrices.

   Justification

   Check factor sides and the full identity relationship from the source row ordering and actual option5.

   Supplementary explanation
4. Expression

   $$C=KP^{-1}$$
   Explanation

   Right multiplication by P⁻¹ reorders columns of K as3,1,2, not its rows.

   Justification

   Check factor sides and the full identity relationship from the source row ordering and actual option5.

   Supplementary explanation
5. Expression

   $$C=\begin{bmatrix}c'&a'&b'\\f'&d'&e'\\i'&g'&h'\end{bmatrix}$$
   Explanation

   This is actual option5; the early model option3 is not the complete source conclusion.

   Justification

   Check factor sides and the full identity relationship from the source row ordering and actual option5.

   Supplementary explanation

Option5: $C=\begin{bmatrix}c'&a'&b'\\f'&d'&e'\\i'&g'&h'\end{bmatrix}$.

Editorial checks give NC=PAKP⁻¹=I₃ and CN=KP⁻¹PA=I₃, verifying every entry rather than only three diagonal1 entries.

1. Formula
   Observation

   The screen fully presents the problem: Given that the inverse of $\begin{bmatrix} a&b&c\\ d&e&f\\ g&h&i \end{bmatrix}$ is $\begin{bmatrix} a'&b'&c'\\ d'&e'&f'\\ g'&h'&i' \end{bmatrix}$, ask for the inverse of $\begin{bmatrix} g&h&i\\ a&b&c\\ d&e&f \end{bmatrix}$, listing five options (1) to (5).

2. Audio
   Observation

   The narration uses the original rows and corresponding inverse columns to identify the inverse after rearrangement.

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

Given that the inverse of $A=\begin{bmatrix} a&b&c\\ d&e&f\\ g&h&i \end{bmatrix}$ is $A^{-1}=\begin{bmatrix} a'&b'&c'\\ d'&e'&f'\\ g'&h'&i' \end{bmatrix}$, find the inverse of $\begin{bmatrix} g&h&i\\ a&b&c\\ d&e&f \end{bmatrix}$.

1. $A^{-1}=\begin{bmatrix} a'&b'&c'\\ d'&e'&f'\\ g'&h'&i' \end{bmatrix}$

2. The matrix to be found is $\begin{bmatrix} g&h&i\\ a&b&c\\ d&e&f \end{bmatrix}$

3. Actual option5 is$\begin{bmatrix}c'&a'&b'\\f'&d'&e'\\i'&g'&h'\end{bmatrix}$

Select the correct inverse matrix from the five candidates.

1. Expression

   $$A A^{-1}=I_3$$
   Explanation

   First use the known inverse matrix to determine the original row-column pairing relationships.

   Justification

   Check factor sides and the full identity relationship from the source row ordering and actual option5.

   Supplementary explanation
2. Expression

   $$[a\ b\ c]\leftrightarrow\begin{bmatrix} a'\\ d'\\ g' \end{bmatrix}$$
   Explanation

   The first row of the original matrix $[a\ b\ c]$ corresponds to the first column $\begin{bmatrix} a'\\ d'\\ g' \end{bmatrix}$ in the original inverse matrix.

   Justification

   Check factor sides and the full identity relationship from the source row ordering and actual option5.

   Supplementary explanation
3. Expression

   $$[g\ h\ i;\ a\ b\ c;\ d\ e\ f]$$
   Explanation

   The matrix to be found is obtained by rearranging the rows of the original matrix.

   Justification

   Check factor sides and the full identity relationship from the source row ordering and actual option5.

   Supplementary explanation
4. Expression

   $$\begin{bmatrix}c'&a'&b'\\f'&d'&e'\\i'&g'&h'\end{bmatrix}$$
   Explanation

   To maintain the same row-column pairing, the inverse matrix must also rearrange its columns accordingly; option (5) meets this requirement.

   Justification

   Check factor sides and the full identity relationship from the source row ordering and actual option5.

   Supplementary explanation

Option5: $C=\begin{bmatrix}c'&a'&b'\\f'&d'&e'\\i'&g'&h'\end{bmatrix}$.

Editorial checks give NC=PAKP⁻¹=I₃ and CN=KP⁻¹PA=I₃, verifying every entry rather than only three diagonal1 entries.

1. Diagram
   Observation

   The speaker circles corresponding rows of A and N with a green pen and uses arrows to indicate the direction of row movement

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

1. Three rows of matrix A

2. Three rows of matrix N

3. Green circles

4. Arrows

1. The first row of A is circled and connected to the second row of N

2. The second row of A is circled and connected to the third row of N

3. The third row of A is circled and connected to the first row of N

1. The matrix elements themselves remain unchanged

2. The internal order of the rows remains unchanged

Visually presents that N is obtained from A via cyclic row permutation, helping to understand the relationship between the two matrices.

1. Diagram
   Observation

   The source gives A and its inverse K and uses the identity criterion; editorial notation is AK=KA=I₃ with explicit row and column orientation.

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

1. Matrix $A^{-1}$

2. Matrix A

3. Equals sign

4. Identity matrix I

1. Writes out the structure of $A^{-1}$$A = I$

1. The elements of A and $A^{-1}$ remain unchanged

Emphasizes the definition of the inverse matrix: the product of the two matrices is the identity matrix, laying the foundation for subsequent derivation.

1. Diagram
   Observation

   The screen uses green, pink, and cyan to mark rows and columns in the original and inverse matrices respectively, connecting corresponding positions with arrows.

2. Animation
   Observation

   As the speaker explains, circles and arrows appear gradually, first marking $[a\ b\ c]$ and $\begin{bmatrix} a'\\ d'\\ g' \end{bmatrix}$, then marking other pairings.

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

1. Original matrix $A$

2. Inverse matrix $A^{-1}$

3. Matrix to be found $\begin{bmatrix} g&h&i\\ a&b&c\\ d&e&f \end{bmatrix}$

4. Options (1) to (5)

1. First circle the first row of the original matrix and the first column of the inverse matrix

2. Then shift focus to the second row $[a\ b\ c]$ of the matrix to be found

3. Next use an arrow to connect it to the middle column of option (5)

4. Finally supplement the other two sets of row-column correspondences

1. The matrix element symbols themselves remain unchanged

2. What changes is the arrangement position of rows and columns

3. The criterion remains that the product must equal the identity matrix

The visual focus is on transforming abstract matrix multiplication into visible "which row pairs with which column," making the correspondence between row swaps and column swaps clear at a glance.

1. Audio
   Observation

   The narration uses the original rows and corresponding inverse columns to identify the inverse after rearrangement.

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

Seeing a inverse matrix problem, the intuition is to use the adjugate matrix or Gaussian elimination to calculate it explicitly.

This problem utilizes the structural characteristics of row permutations. One only needs to analyze the permutation relationship between the matrices, without performing complex inverse matrix calculations.

1. Audio
   Observation

   The narration uses the original rows and corresponding inverse columns to identify the inverse after rearrangement.

2. Diagram
   Observation

   Multiple options contain identical symbols, but only the option with correct column positions passes the verification.

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

Seeing the same symbols like $a',b',c',\dots$ in the candidate matrix and assuming it must be the inverse matrix.

The key to an inverse matrix is not having the same set of elements, but whether multiplying it with the original matrix yields the identity matrix; therefore, the correspondence positions of rows and columns must be checked.

1. Audio
   Observation

   The narration uses the original rows and corresponding inverse columns to identify the inverse after rearrangement.

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

After the rows of the original matrix are rearranged, still using the original column order of the inverse matrix for pairing.

Once the row positions of the left matrix change, the corresponding column positions of the right matrix must also change, otherwise the product will not be the identity matrix.

1. Audio
   Observation

   The narration uses the original rows and corresponding inverse columns to identify the inverse after rearrangement.

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

When deriving the inverse of N, the core basis is the definition of the inverse matrix $A^{-1}$$A=I$, and the derived multiplicative property $(PA)^{-1}$=$A^{-1}P^{-1}$.

1. Audio
   Observation

   The narration uses the original rows and corresponding inverse columns to identify the inverse after rearrangement.

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

The concept of row permutation is applied to express N as PA, and then the properties of inverse matrices are used to find $N^{-1}$.

1. Audio
   Observation

   The narration uses the original rows and corresponding inverse columns to identify the inverse after rearrangement.

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

A matrix multiplied by its inverse gives the identity. Row-column inner products implement that criterion for checking a candidate inverse.

1. Audio
   Observation

   The narration uses the original rows and corresponding inverse columns to identify the inverse after rearrangement.

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

From single row-column pairing checks, generalize to a more common problem-solving rule: when rows of the left matrix are swapped, columns of the right matrix must be swapped.

1. Audio
   Observation

   The narration uses the original rows and corresponding inverse columns to identify the inverse after rearrangement.

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

1. Definition of Inverse Matrix
2. Row Permutation of Matrices
3. Derivation of the Inverse of N
4. Matrix inverse row-rearrangement exercise (101, question4)

1. Audio
   Observation

   The narration uses the original rows and corresponding inverse columns to identify the inverse after rearrangement.

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

1. Misconception that Hard Calculation is Required
2. Row Permutation of Matrices

1. Audio
   Observation

   The narration uses the original rows and corresponding inverse columns to identify the inverse after rearrangement.

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

1. Product of Inverse and Original Matrix Yields Identity Matrix
2. "Row-Column Pairing" Rule for Matrix Multiplication
3. Row Swap in Original Matrix Corresponds to Column Swap in Inverse Matrix

1. Formula
   Observation

   Editorial notation writes the source identity criterion as NC=I₃ when checking candidate C.

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

1. Product of Inverse and Original Matrix Yields Identity Matrix
2. "Row-Column Pairing" Rule for Matrix Multiplication

1. Audio
   Observation

   The narration uses the original rows and corresponding inverse columns to identify the inverse after rearrangement.

1. Editorial checks use actual final option5, the given matrices and independent permutation multiplication. Early model option3 and column transcription errors are corrected without attributing an author error.

1. Deducing Option (5) from Row Position Change
2. Re-verifying Option (5) Using Remaining Row-Column Pairings
3. Finding the Inverse After Row Rearrangement