Question

M is a 2 x 2 matrix with eigenvalues 4 and 9 the eigen values of $M^2$ are

• A. -2 and -3
• B. 2 and 3
• C. 4 and 9
• D. 16 and 81

Answer 1

The correct option is D 16 and 81

When M is required, the eigen vectors remain unchanged whereas, the eigen values are squared

Hence, eigen values of $M^2$ are 16 and 81.

Proof: let $\lambda$ be the eigen values of $A^2$, then

$|A^2-\lambda I|$ = 0

or $(|A-\sqrt{\lambda I}|)(|A+\sqrt{\lambda I}|))=0Hence,either|A-\sqrt{\lambda }I|$ = 0

or $|A+\sqrt{\lambda }I|$ = 0

Eigen values of A are $\sqrt{\lambda }$ or -$\sqrt{\lambda }$