If A is a singular square matrix of order 3, then \(A\,(\mathrm{adj}\,A)\) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → the null matrix O
For every square matrix A, the product with its adjoint is
\(A\,(\mathrm{adj}\,A) = |A|\,I\)
A singular matrix has \(|A| = 0\):
\(A\,(\mathrm{adj}\,A) = 0 \cdot I = O\)
For example, for \(A = \begin{bmatrix}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 0\end{bmatrix}\), \(\mathrm{adj}\,A = \begin{bmatrix}0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 1\end{bmatrix}\) and the product is O.