Question

If $A=\left[\begin{array}{cc}\lambda \mu & {\mu }^2\\ -{\lambda }^2& -\lambda \mu \end{array}\right]$A=[λμμ2−λ2−λμ] then $A^2$ is

• A. an idempotent matrix
• B. a nilpotent matrix
• C. an orthogonal matrix
• D. a symmetric matrix

Answer 1

The correct option is B a nilpotent matrix

Given $A=\left[\begin{array}{cc}\lambda \mu & {\mu }^2\\ -{\lambda }^2& -\lambda \mu \end{array}\right]$

$A^2=\left[\begin{array}{cc}\lambda \mu & {\mu }^2\\ -{\lambda }^2& -\lambda \mu \end{array}\right]\left[\begin{array}{cc}\lambda \mu & {\mu }^2\\ -{\lambda }^2& -\lambda \mu \end{array}\right]$

$=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

Hence, $A^2=0$

The matrix $A$ is said to be Nilpotent if $A_m=0$ where $m$ is an integer and $0$ is a null matrix of the same order as of $A.$

Hence, $A^2$ is a nilpotent matrix.