Question

If A and B are two square matrices of order `3xx3` which satify `AB=A` and `BA=B`, then
Which of the following is true ?
A. If matrix A is singular, then matrix B is nonsingular.
B. If matrix A is nonsingular, then materix B is singular.
C. If matrix A is singular, then matrix B is also singular.
D. Cannot say anything.

Answer 1

Correct Answer - C
`AB=A implies |AB|=|A|` (1)
`implies |A|=0` or `|B|=1`
`BA=B implies |BA|=|B|` (2)
`implies |A|=1` or `|B|=0`
If `|A|=0`, then from Eq. (2) `|B|=0`
If `|B|=0`, then from Eq. (1), `|A|=0`