Matrices & Determinants
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: Apply the multiplicative property of determinants, $|AB| = |A||B|$, to transform matrix product equations into scalar algebraic equations involving $|A|$ and $|B|$.
$AB = A \Rightarrow |AB| = |A| \Rightarrow |A||B| = |A| \Rightarrow |A| = 0$ 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