Why is the answer to the matrix inverse row-rearrangement problem option (5)?

Conditions

• The original matrix $A$ has inverse $A^{-1}$.
• The new matrix $N$ is formed by permuting the rows of $A$ as 3, 1, 2.
• The candidate matrices contain the elements of $A^{-1}$ in various orders.

Reasoning, step by step

1. Analyze the row permutation of $N$ relative to $A$: row 3 becomes row 1, row 1 becomes row 2, row 2 becomes row 3.
2. Determine the required column permutation for the inverse: column 3 of $A^{-1}$ must become column 1, column 1 must become column 2, column 2 must become column 3.
3. Examine the given options to find the one that matches this column ordering.
4. Verify that the chosen option satisfies the row-column pairing for the identity matrix.

Example

The rows of $N$ are $[g\ h\ i]$, $[a\ b\ c]$, $[d\ e\ f]$. The columns of $A^{-1}$ are $\begin{bmatrix} a' \\ d' \\ g' \end{bmatrix}$, $\begin{bmatrix} b' \\ e' \\ h' \end{bmatrix}$, $\begin{bmatrix} c' \\ f' \\ i' \end{bmatrix}$. Option (5) is $\begin{bmatrix} c' & a' & b' \\ f' & d' & e' \\ i' & g' & h' \end{bmatrix}$. Its columns are $\begin{bmatrix} c' \\ f' \\ i' \end{bmatrix}$, $\begin{bmatrix} a' \\ d' \\ g' \end{bmatrix}$, $\begin{bmatrix} b' \\ e' \\ h' \end{bmatrix}$. Pairing row 1 of $N$ with col 1 of option (5) gives $[g\ h\ i] \cdot \begin{bmatrix} c' \\ f' \\ i' \end{bmatrix} = gc' + hf' + ii' = 1$ (from $A A^{-1}=I$). Pairing row 2 of $N$ with col 2 gives $[a\ b\ c] \cdot \begin{bmatrix} a' \\ d' \\ g' \end{bmatrix} = aa' + bb' + cc' = 1$. All pairings match the identity matrix.

Common misconceptions

• Selecting an option that has the correct elements but in the wrong column order.
• Assuming that only the diagonal entries need to be 1, without checking that the off-diagonal entries are 0.

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Connected concepts

Explore next

• How to quickly verify if a matrix is the inverse of another using row-column pairing?
• What is the permutation matrix representation of this row rearrangement?
• Why does swapping rows in the original matrix require swapping columns in the inverse matrix?

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