Question

If A and B are non-zero square matices, then AB=$0$ implies

• A. A and B are othogonal
• B. A and B are singular
• C. B is singular
• D. A is singular

Answer 1

The correct option is B A and B are singular

AB=0
$\left|AB\right|=\left|0\right|$
$\left|A\right|\left|B\right|=0$
This implies that either
$\left|A\right|=0or\left|B\right|=0orboth\left|A\right|and\left|B\right|is0$
Hence option (d) is incorrect now lets assume A is sngular and B is non singular
$\Rightarrow \left|B\right|\ne 0means{B}^{-1}exist$
AB=0
$AB{B}^{-1}=O{B}^{-1}$
A=O
This is the contradiction with the given statement similarly, if B is singular and A is non singular then
$\Rightarrow \left|A\right|\ne 0means{A}^{-1}exist$
AB=O
$A^{-1}AB=A^{-1}O$
This is also contradiction with the given statement hence if A and B are non zero matrices then AB=O is only possible when determinant of A and determinant of B is 0,
Hence option (b) is correct .