If \(A = \begin{bmatrix}2&-1\\-4&2\end{bmatrix}\), \(B = \begin{bmatrix}1&k\\2&6\end{bmatrix}\) and \(AB = O\), where O is the zero matrix of order 2, then the value of \(k\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 3
The element in row i, column j of AB is row i of A times column j of B. The first column of B gives
\(2(1) + (-1)(2) = 0\)
\(-4(1) + 2(2) = 0\)
so the first column of AB is already zero. The second column of B gives
\(2k + (-1)(6) = 2k - 6\)
\(-4k + 2(6) = -4k + 12\)
Both must be 0:
\(2k - 6 = 0\)
\(k = 3\)
k = 3 also makes \(-4k + 12 = 0\). Note that AB = O here although neither A nor B is the zero matrix.