Matrices & Determinants
General
Grade 12
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?
If matrix $A$ is singular then matrix $B$ is non-singular
If matrix $A$ is non-singular then matrix $B$ is singular
If matrix $A$ is singular then matrix $B$ is also singular
Cannot say anything.
Step-by-Step Solution
Key Concept: General
$AB = A \Rightarrow |AB| = |A| \Rightarrow |A||B| = |A| \Rightarrow |A| = 0$ or $|B| = 1$ ...(1)<br>$BA = B \Rightarrow |BA| = |B| \Rightarrow |B||A| = |B| \Rightarrow |A| = 1$ or $|B| = 0$ ...(2)<br>If $|A| = 0$, then from Eq. (2), $|B| = 0$<br>If $|B| = 0$, then from Eq. (1), $|A| = 0$
Correct Answer: C