Discussion
Let \(A\) be an invertible \(5 \times 5\) real matrix. Let \(B\) be the matrix obtained by interchanging the second and third columns of \(A\) and then adding 3 times the third column to the fifth column.
Which ONE of the following statements is TRUE?
Answer: B
\(B^{-1}\) is obtained by interchanging the second and third rows of \(A^{-1}\), and then adding \(-3\) times the fifth row to the third row.
Column operations are right multiplication by elementary matrices. Let \(P\) swap columns 2 and 3, and \(E = I + 3e_3e_5^{T}\) add 3 times column 3 to column 5. Then
\[ \begin{aligned} B &= APE \\ B^{-1} &= E^{-1}P^{-1}A^{-1} \\ &= (I - 3e_3e_5^{T})\,P\,A^{-1} \end{aligned} \]
Left multiplication is a row operation, applied from the right-hand factor outwards: first \(P\) swaps rows 2 and 3 of \(A^{-1}\); then \(I - 3e_3e_5^{T}\) replaces row 3 by row 3 minus 3 times row 5.
The first option gets the swap right but carries the column operation over unchanged. Inverting it flips the sign to \(-3\), and moving from columns to rows exchanges the roles of the third and fifth lines. The two options that add before swapping have the order reversed: inverting a product reverses the order of its factors.
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