Question

If $A$ and $B$ are two square matrices of order $3\times 3$ which satisfy $AB=A$ and $BA=B$, then Which of the following is true?

• A. If matrix $A$ is singular, then matrix $B$ is non singular.
• B. If matrix $A$ is nonsingular, then matrix $B$ is singular.
• C. If matrix $A$ is singular, then matrix $B$ is also singular.
• D. Cannot say anything.

Answer 1

The correct option is C If matrix $A$ is singular, then matrix $B$ is also singular.

$A$ and $B$ are two square matrices of order $3\times 3$ which satisfy $AB=A$ and $BA=B,$ then if matrix $A$ is singular, then matrix $B$ is also singular .

$A$ singular matrix is that matrix whose determinants is zero and which is non-irreversible.

$\therefore A$ and $B$ both are singular matrices, then only the conditions $AB=A$ and $BA=B$ holds true.

Hence, the answer is if matrix $A$ is singular, then matrix $B$ is also singular.