Q68 / 180 · COMEDK 2021 · Mathematics

If matrix $A=\left[\begin{array}{cc}2 & -2 \\ -2 & 2\end{array}\right]$ and $A^{2}=p A$, then the value of $p$ is

  • A
    6
  • B
    $-4$
  • Correct answer
    C
    4
  • D
    8

Solution

Given $A = \begin{bmatrix} 2 & -2 \\ -2 & 2 \end{bmatrix}$.

Calculate $A^{2}$ by multiplying $A$ with itself:

$A^{2} = \begin{bmatrix} 2 & -2 \\ -2 & 2 \end{bmatrix} \begin{bmatrix} 2 & -2 \\ -2 & 2 \end{bmatrix}$

$A^{2} = \begin{bmatrix} (2)(2) + (-2)(-2) & (2)(-2) + (-2)(2) \\ (-2)(2) + (2)(-2) & (-2)(-2) + (2)(2) \end{bmatrix}$

$A^{2} = \begin{bmatrix} 4 + 4 & -4 - 4 \\ -4 - 4 & 4 + 4 \end{bmatrix} = \begin{bmatrix} 8 & -8 \\ -8 & 8 \end{bmatrix}$

We are given $A^{2} = pA$. Substituting the matrices:

$\begin{bmatrix} 8 & -8 \\ -8 & 8 \end{bmatrix} = p \begin{bmatrix} 2 & -2 \\ -2 & 2 \end{bmatrix}$

$\begin{bmatrix} 8 & -8 \\ -8 & 8 \end{bmatrix} = \begin{bmatrix} 2p & -2p \\ -2p & 2p \end{bmatrix}$

Comparing the corresponding elements, $2p = 8$, which implies $p = 4$.

Answer: 4

Hence, the correct answer is option: 'C'.

Relevant Questions