1. Tardigrade
  2. Question
  3. Mathematics
  4. If A =[2 -1 -1 2] and I is the unit matrix of order 2 , then A 2 equals

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. If $A =\begin{bmatrix}2 & -1 \\ -1 & 2\end{bmatrix}$ and $I$ is the unit matrix of order $2$ , then $A ^{2}$ equals

BITSATBITSAT 2007

Solution:

$A=\begin{bmatrix}2 & -1 \\ -1 & 2\end{bmatrix}$
$A ^{2}= A \times A =\begin{bmatrix}2 & -1 \\ -1 & 2\end{bmatrix} \times\begin{bmatrix}2 & -1 \\ -1 & 2\end{bmatrix}$
$=\begin{bmatrix}4+1 & -2-2 \\ -2-2 & 1+4\end{bmatrix}=\begin{bmatrix}5 & -4 \\ -4 & 5\end{bmatrix}$
From the given option to get first entry i.e $5$ of $A ^{2}$,
only option (A) is satisfied. Or we can check as below
$4 A -3 I =\begin{bmatrix}8 & -4 \\ -4 & 8\end{bmatrix}-3 \times\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$
$=\begin{bmatrix}8 & -4 \\ -4 & 8\end{bmatrix}-\begin{bmatrix}3 & 0 \\ 0 & 3\end{bmatrix}$
$=\begin{bmatrix}5 & -4 \\ -4 & 5\end{bmatrix}= A ^{2}$