Question

Say true or false:

If A and B are square matrices of order n, then A and B will commute, if $|A-\lambda |$ and $|B-\lambda |$ commute for every scalar $\lambda$.

• A. True
• B. False

Answer 1

The correct option is A True

$(A-\left|\lambda \right|)$ and $(B-\left|\lambda \right|)$ commute then
$(A-\left|\lambda \right|)(B-\left|\lambda \right|)=(B-\left|\lambda \right|)(A-\left|\lambda \right|)\Rightarrow AB-\left|\lambda \right|B-A\left|\lambda \right|+{\lambda }^2=BA-B\left|\lambda \right|-\left|\lambda \right|A+{\lambda }^2$
Hence $AB=BA$ only when
$\left|\lambda \right|B=B\left|\lambda \right|$ and $\left|\lambda \right|A=A\left|\lambda \right|$