Question

A and B are two non-singular square matrices of order $3\times 3$ such that AB = A and BA = B then

• A. $|A-B|=6$
• B. $|A+B|=0$
• C. $|A-B{|}^2=0$
• D. $|A+B|=8$

Answer 1

Answer: (B), (C), (D)

The correct options are
B $|A+B|=0$
C $|A-B{|}^2=0$
D $|A+B|=8$

$\begin{array}{cc}& \\ AB=A,BA=B& (A+B{)}^2=A^2+B^2+AB+BA\\ \Rightarrow A^2=AA& =2(A+B)\\ =ABA=AB=A& |A+B{|}^2=8|A+B|\\ Similarly{B}^2=B& \Rightarrow |A+B|(|A+B|-8)=0\\ & \Rightarrow |A+B|=0or\Rightarrow |A+B|=8\\ & Similarly|A+B{|}^2=0\end{array}$