Recovering a Matrix and Swapping Its Columns

1. Formula
   Observation

   The problem text on screen states 'Let M be $a 3 \times 2$ matrix', and later handwriting shows M = [3 1; 2 2; 1 3].

1. Editorial orientation terms clarify regional usage; source values, factor order and timestamps are retained.

M

An unknown $3\times 2$ matrix, which is the main subject required for derivation and application in the problem.

$3\times 2$ matrix

1. Formula
   Observation

   The right-multiplied matrix in the first condition is [1 0; 0 0].

2. Audio
   Observation

   The narration relates the current matrix equation to the extraction, reconstruction or swapping of vertical columns; notes paraphrase it with explicit orientation.

1. Editorial orientation terms clarify regional usage; source values, factor order and timestamps are retained.

[1 0; 0 0]

$A 2\times 2$ matrix acting on the right side of M; in the context of this problem, it is equivalent to an operation that preserves the first column of M and zeros out the second column.

$2\times 2$ matrix

1. Formula
   Observation

   The right-multiplied matrix in the second condition is [0 0; 0 1].

2. Audio
   Observation

   The narration relates the current matrix equation to the extraction, reconstruction or swapping of vertical columns; notes paraphrase it with explicit orientation.

1. Editorial orientation terms clarify regional usage; source values, factor order and timestamps are retained.

[0 0; 0 1]

$A 2\times 2$ matrix acting on the right side of M; in the context of this problem, it is equivalent to an operation that preserves the second column of M and zeros out the first column.

$2\times 2$ matrix

1. Formula
   Observation

   The problem finally asks to calculate M[0 1; 1 0].

1. Editorial orientation terms clarify regional usage; source values, factor order and timestamps are retained.

[0 1; 1 0]

$A 2\times 2$ matrix acting on the right side of M; in the context of this problem, it corresponds to swapping the two columns of M.

$2\times 2$ matrix

1. Formula
   Observation

   The first column of the resulting matrix from the first condition is 3, 2, 1.

2. Audio
   Observation

   The narration relates the current matrix equation to the extraction, reconstruction or swapping of vertical columns; notes paraphrase it with explicit orientation.

1. Editorial orientation terms clarify regional usage; source values, factor order and timestamps are retained.

(3,2,1)^T

The first column of the resulting matrix obtained from M[1 0; 0 0], which equals the first column of M.

Column vector in $R^3$

1. Formula
   Observation

   The second column of the resulting matrix from the second condition is 1, 2, 3.

2. Audio
   Observation

   The narration relates the current matrix equation to the extraction, reconstruction or swapping of vertical columns; notes paraphrase it with explicit orientation.

1. Editorial orientation terms clarify regional usage; source values, factor order and timestamps are retained.

(1,2,3)^T

The second column of the resulting matrix obtained from M[0 0; 0 1], which equals the second column of M.

Column vector in $R^3$

1. Audio
   Observation

   The narration relates the current matrix equation to the extraction, reconstruction or swapping of vertical columns; notes paraphrase it with explicit orientation.

2. Formula
   Observation

   All three conditions involve M being right-multiplied by $a 2\times 2$ matrix, with results showing extraction of the first column, extraction of the second column, and swapping of columns, respectively.

1. Editorial orientation terms clarify regional usage; source values, factor order and timestamps are retained.

The core method of this problem is to view the $2\times 2$ matrix on the right side of M as performing linear combinations on the columns of M. Right multiplying by [1 0; 0 0] preserves the first column and zeros out the second column; right multiplying by [0 0; 0 1] preserves the second column and zeros out the first column; right multiplying by [0 1; 1 0] swaps the two columns.

1. M is $a 3\times 2$ matrix

2. The right-multiplied matrix is $2\times 2$

3. The result is still $a 3\times 2$ matrix

1. Formula
   Observation

   The screen first writes M = [3 1; 2 2; 1 3].

2. Audio
   Observation

   The narration relates the current matrix equation to the extraction, reconstruction or swapping of vertical columns; notes paraphrase it with explicit orientation.

1. Editorial orientation terms clarify regional usage; source values, factor order and timestamps are retained.

From the result of M[1 0; 0 0], we know the first column of M is (3,2,1). From the result of M[0 0; 0 1], we know the second column of M is (1,2,3). Placing these two columns back into their original positions yields M = [3 1; 2 2; 1 3].

1. The results of M[1 0; 0 0] and M[0 0; 0 1] are known

2. M is $a 3\times 2$ matrix

1. Right multiplication by $a 2\times 2$ matrix corresponds to column operations

1. Formula
   Observation

   The problem requires calculating M[0 1; 1 0].

2. Audio
   Observation

   The narration relates the current matrix equation to the extraction, reconstruction or swapping of vertical columns; notes paraphrase it with explicit orientation.

1. Editorial orientation terms clarify regional usage; source values, factor order and timestamps are retained.

Once we know the two columns of M are (3,2,1) and (1,2,3) respectively, right multiplying by [0 1; 1 0] swaps the first and second columns. Therefore, the resulting first column becomes (1,2,3), and the second column becomes (3,2,1).

1. The two columns of M have been determined

2. The right-multiplied matrix is [0 1; 1 0]

1. Reconstructing M from two conditions

1. Audio
   Observation

   The narration relates the current matrix equation to the extraction, reconstruction or swapping of vertical columns; notes paraphrase it with explicit orientation.

2. Formula
   Observation

   All three examples show that the $2\times 2$ matrix on the right changes the arrangement or preservation of M's columns.

1. Editorial orientation terms clarify regional usage; source values, factor order and timestamps are retained.

For the3×2 matrix M and $a2\times 2$ right factor, each result column is a linear combination of the two columns of M. The three source factors retain the first column, retain the second column or swap both columns; the general formula is an editorial explanation.

1. M is $a 3\times 2$ matrix

2. The right-multiplied matrix is $2\times 2$

Holds for all three right-multiplied matrices appearing in this problem.

1. Formula
   Observation

   The screen sequentially marks the column operations corresponding to [1 0; 0 0], [0 0; 0 1], and [0 1; 1 0].

2. Audio
   Observation

   The narration relates the current matrix equation to the extraction, reconstruction or swapping of vertical columns; notes paraphrase it with explicit orientation.

1. Editorial orientation terms clarify regional usage; source values, factor order and timestamps are retained.

1. Expression

   $$M[1 0; 0 0] = [3 0; 2 0; 1 0]$$
   Explanation

   The first condition tells us that after right multiplying by [1 0; 0 0], the first column obtained is the first column of M.

   Justification

   Compare the actual problem equations by vertical columns; each column of the right factor gives the corresponding combination coefficients.

   Supplementary explanation
2. Expression

   $$M_{:,1}=(3,2,1)^T$$
   Explanation

   Comparing the first column of the resulting matrix with the column structure of M, we find that the first column of M is 3, 2, 1.

   Justification

   Compare the actual problem equations by vertical columns; each column of the right factor gives the corresponding combination coefficients.

   Supplementary explanation
3. Expression

   $$M[0 0; 0 1] = [0 1; 0 2; 0 3]$$
   Explanation

   The second condition tells us that after right multiplying by [0 0; 0 1], the second column obtained is the second column of M.

   Justification

   Compare the actual problem equations by vertical columns; each column of the right factor gives the corresponding combination coefficients.

   Supplementary explanation
4. Expression

   $$M_{:,2}=(1,2,3)^T$$
   Explanation

   Comparing the second column of the resulting matrix with the column structure of M, we find that the second column of M is 1, 2, 3.

   Justification

   Compare the actual problem equations by vertical columns; each column of the right factor gives the corresponding combination coefficients.

   Supplementary explanation
5. Expression

   $$M = [3 1; 2 2; 1 3]$$
   Explanation

   Placing the two columns just found back into their original positions reconstructs the complete $3\times 2$ matrix M.

   Justification

   Compare the actual problem equations by vertical columns; each column of the right factor gives the corresponding combination coefficients.

   Supplementary explanation
6. Expression

   $$M[0 1; 1 0] = [1 3; 2 2; 3 1]$$
   Explanation

   Right multiplying by [0 1; 1 0] swaps the two columns of M, so the first column becomes (1,2,3) and the second column becomes (3,2,1).

   Justification

   Compare the actual problem equations by vertical columns; each column of the right factor gives the corresponding combination coefficients.

   Supplementary explanation

The final answer is [1 3; 2 2; 3 1].

1. Formula
   Observation

   The screen fully presents the problem: Let M be $a 3\times 2$ matrix, such that M[1 0; 0 0]=[3 0; 2 0; 1 0] and M[0 0; 0 1]=[0 1; 0 2; 0 3]. Find M[0 1; 1 0].

2. Audio
   Observation

   The narration relates the current matrix equation to the extraction, reconstruction or swapping of vertical columns; notes paraphrase it with explicit orientation.

1. Editorial orientation terms clarify regional usage; source values, factor order and timestamps are retained.

Let M be $a 3\times 2$ matrix, and suppose M[1 0; 0 0]=[3 0; 2 0; 1 0] and M[0 0; 0 1]=[0 1; 0 2; 0 3]. Then M[0 1; 1 0]=______.

1. M is $a 3\times 2$ matrix

2. M[1 0; 0 0]=[3 0; 2 0; 1 0]

3. M[0 0; 0 1]=[0 1; 0 2; 0 3]

Find the value of M[0 1; 1 0].

1. Expression

   $$M[1 0; 0 0]=[3 0; 2 0; 1 0]$$
   Explanation

   First understand the first condition: right multiplying by [1 0; 0 0] extracts the first column of M.

   Justification

   Compare the actual problem equations by vertical columns; each column of the right factor gives the corresponding combination coefficients.

   Supplementary explanation
2. Expression

   $$M_{:,1}=(3,2,1)^T$$
   Explanation

   Therefore, the first column of M is the first column of the resulting matrix.

   Justification

   Compare the actual problem equations by vertical columns; each column of the right factor gives the corresponding combination coefficients.

   Supplementary explanation
3. Expression

   $$M[0 0; 0 1]=[0 1; 0 2; 0 3]$$
   Explanation

   Next understand the second condition: right multiplying by [0 0; 0 1] extracts the second column of M.

   Justification

   Compare the actual problem equations by vertical columns; each column of the right factor gives the corresponding combination coefficients.

   Supplementary explanation
4. Expression

   $$M_{:,2}=(1,2,3)^T$$
   Explanation

   Therefore, the second column of M is the second column of the resulting matrix.

   Justification

   Compare the actual problem equations by vertical columns; each column of the right factor gives the corresponding combination coefficients.

   Supplementary explanation
5. Expression

   $$M=[3 1; 2 2; 1 3]$$
   Explanation

   Combine the two columns to reconstruct M.

   Justification

   Compare the actual problem equations by vertical columns; each column of the right factor gives the corresponding combination coefficients.

   Supplementary explanation
6. Expression

   $$M[0 1; 1 0]=[1 3; 2 2; 3 1]$$
   Explanation

   Right multiplying by [0 1; 1 0] swaps the two columns of M, yielding the final answer.

   Justification

   Compare the actual problem equations by vertical columns; each column of the right factor gives the corresponding combination coefficients.

   Supplementary explanation

[1 3; 2 2; 3 1]

Substituting the determined M back into the two known conditions indeed yields [3 0; 2 0; 1 0] and [0 1; 0 2; 0 3] respectively. Swapping the two columns then gives [1 3; 2 2; 3 1].

1. Animation
   Observation

   The screen circles the first column of [1 0; 0 0] in green and circles the first column 3, 2, 1 of the resulting matrix.

2. Audio
   Observation

   The narration relates the current matrix equation to the extraction, reconstruction or swapping of vertical columns; notes paraphrase it with explicit orientation.

1. Editorial orientation terms clarify regional usage; source values, factor order and timestamps are retained.

1. Right-multiplied matrix [1 0; 0 0]

2. Resulting matrix [3 0; 2 0; 1 0]

3. First column of M

1. Green circle selection moves from the right-multiplied matrix to the first column of the resulting matrix

2. Handwritten annotation C1 points to the first column

1. M remains $a 3\times 2$ matrix

2. The second column of the resulting matrix remains 0

Visually emphasizes that right multiplying by [1 0; 0 0] only preserves the first column of M, while the second column is zeroed out.

1. Animation
   Observation

   The screen circles the second column of [0 0; 0 1] in green and circles the second column 1, 2, 3 of the resulting matrix.

2. Audio
   Observation

   The narration relates the current matrix equation to the extraction, reconstruction or swapping of vertical columns; notes paraphrase it with explicit orientation.

1. Editorial orientation terms clarify regional usage; source values, factor order and timestamps are retained.

1. Right-multiplied matrix [0 0; 0 1]

2. Resulting matrix [0 1; 0 2; 0 3]

3. Second column of M

1. Green circle selection shifts to the second condition

2. Handwritten annotation C2 points to the second column

1. The first column of the resulting matrix remains 0

Visually emphasizes that right multiplying by [0 0; 0 1] only preserves the second column of M, while the first column is zeroed out.

1. Animation
   Observation

   The bottom left of the screen gradually handwrites M = [3 1; 2 2; 1 3].

1. Editorial orientation terms clarify regional usage; source values, factor order and timestamps are retained.

1. Writing process of M

2. First column (3,2,1)

3. Second column (1,2,3)

1. First writes the first column 3, 2, 1

2. Then adds the second column 1, 2, 3

1. The dimension of M is always $3\times 2$

Combining the column information given by the two conditions into a complete matrix.

1. Animation
   Observation

   The screen writes the final matrix [1 3; 2 2; 3 1] on the right side of the problem and marks it with a green box.

2. Audio
   Observation

   The narration relates the current matrix equation to the extraction, reconstruction or swapping of vertical columns; notes paraphrase it with explicit orientation.

1. Editorial orientation terms clarify regional usage; source values, factor order and timestamps are retained.

1. Target product M[0 1; 1 0]

2. Final matrix [1 3; 2 2; 3 1]

1. First column changes from (3,2,1) to (1,2,3)

2. Second column changes from (1,2,3) to (3,2,1)

1. Matrix dimension remains $3\times 2$

2. The set of contents of the two columns remains unchanged, only the order is swapped

Visually directly demonstrates that the effect of right multiplying by [0 1; 1 0] is to swap the two columns of M.

1. Audio
   Observation

   The narration relates the current matrix equation to the extraction, reconstruction or swapping of vertical columns; notes paraphrase it with explicit orientation.

2. Formula
   Observation

   The screen circles the columns of the $2\times 2$ matrix on the right and maps them one-to-one to the columns of M.

1. Editorial orientation terms clarify regional usage; source values, factor order and timestamps are retained.

Interpreting the source regional term as a horizontal row operation.

The actual source calculation is correct: $a2\times 2$ right factor combines the two columns of M, while a compatible left factor combines horizontal rows. Regional labels differ; notes specify orientation without attributing a mathematical error to the author.

1. Audio
   Observation

   The narration relates the current matrix equation to the extraction, reconstruction or swapping of vertical columns; notes paraphrase it with explicit orientation.

2. Formula
   Observation

   The three right-multiplied matrices in the example correspond exactly to three types of column operations.

1. Editorial orientation terms clarify regional usage; source values, factor order and timestamps are retained.

This example is a direct application of the method 'right multiplication by $a 2\times 2$ matrix corresponds to column operations'.

1. Formula
   Observation

   First determine M from the two conditions, then calculate M[0 1; 1 0].

1. Editorial orientation terms clarify regional usage; source values, factor order and timestamps are retained.

One must first reconstruct the two columns of M before further determining the result of swapping after right multiplying by [0 1; 1 0].

1. Audio
   Observation

   The narration relates the current matrix equation to the extraction, reconstruction or swapping of vertical columns; notes paraphrase it with explicit orientation.

1. Editorial orientation terms clarify regional usage; source values, factor order and timestamps are retained.

1. Right multiplication by $a 2\times 2$ matrix corresponds to column operations
2. Right multiplication by $a 2\times 2$ matrix performs column operations

1. Formula
   Observation

   The screen derives M=[3 1; 2 2; 1 3] from the two conditions.

1. Editorial orientation terms clarify regional usage; source values, factor order and timestamps are retained.

1. Reconstructing M from two conditions
2. Deriving M from condition matrices and finding the target matrix

1. Formula
   Observation

   The final answer is [1 3; 2 2; 3 1].

2. Audio
   Observation

   The narration relates the current matrix equation to the extraction, reconstruction or swapping of vertical columns; notes paraphrase it with explicit orientation.

1. Editorial orientation terms clarify regional usage; source values, factor order and timestamps are retained.

1. Right multiplying by [0 1; 1 0] represents swapping two columns
2. 090 Fill-in-the-blank 6: column operations in $3\times 2$ matrix multiplication