Question

If $A\neq B, \, AB=BA$ and $A^{2}=B^{2},$ then the value of the determinant of matrix $A+B$ is (where $A$ and $B$ are square matrices of order $3\times 3$ )

• A. $0$
• B. $1$
• C. $3^{3}$
• D. $3^{2}$

Answer 1

Answer: (A)

$A^{2}-B^{2}=\left(A - B\right)\left(A + B\right)=0$ $\left(given A B = B A\right)$
Since, $A\neq B\Rightarrow A-B$ is not a null matrix
Hence, $A+B$ is a null matrix or $det\left(A - B\right)=det\left(A + B\right)=0$
In both cases $det\left(A + B\right)=0$