If \(B = \begin{bmatrix}1 & -2 & k\\ 0 & 1 & -3\\ 0 & 0 & 1\end{bmatrix}\) is the inverse of \(A = \begin{bmatrix}1 & 2 & 0\\ 0 & 1 & 3\\ 0 & 0 & 1\end{bmatrix}\), then the value of k is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 6
If B is the inverse of A, then \(AB = I\). So every entry of AB must match the identity matrix.
The (1, 3) entry of AB is row 1 of A times column 3 of B, and it must be 0:
\(1 \cdot k + 2 \cdot (-3) + 0 \cdot 1 = 0\)
\(k - 6 = 0\)
\(k = 6\)
Check: with k = 6,
\(\displaystyle AB = \begin{bmatrix}1 & -2 + 2 & 6 - 6\\ 0 & 1 & -3 + 3\\ 0 & 0 & 1\end{bmatrix} = I\)