# Let A be a matrix such that $A\cdot \begin{bmatrix} 1 &2 \\ 0 & 3 \end{bmatrix}$  a scalar matrix and $\left | 3A \right |=108$. Then A2 equals :  Option 1) $\begin{bmatrix} 4 &-32 \\ 0 & 36 \end{bmatrix}$       Option 2) $\begin{bmatrix} 36 &0 \\ -32 & 4 \end{bmatrix}$ Option 3) $\begin{bmatrix} 4 &0 \\ -32 & 36 \end{bmatrix}$ Option 4) $\begin{bmatrix} 36 &-32 \\ 0& 4 \end{bmatrix}$

N neha
H Himanshu

As we learned

Scalar Matrix -

A diagonal matrix whose all the elements are equal is called a scalar matrix

- wherein

$A\cdot \begin{bmatrix} 1 &2 \\ 0& 3 \end{bmatrix}$    is a scalar matrix

$\left | 3A \right |=108$

Let scalar matrix be $\begin{bmatrix} k & 0\\ 0 &k \end{bmatrix}$

$A\begin{bmatrix} 1 &2 \\ 0&3 \end{bmatrix}=\begin{bmatrix} k & 0\\ 0 &k \end{bmatrix}$

$A=\begin{bmatrix} k & 0\\ 0 &k \end{bmatrix}\begin{bmatrix} 1 &2 \\ 0&3 \end{bmatrix}^{-1}$

$A=\begin{bmatrix} k & 0\\ 0 &k \end{bmatrix}\begin{bmatrix} 1 &\frac{-2}{3} \\ 0&\frac{1}{3} \end{bmatrix}$

$A=\begin{bmatrix} k &- \frac{2}{3}k\\ 0 &\frac{k}{3} \end{bmatrix}$   also $\left | 3A \right |=108$

$\Rightarrow 3k^{2}=108\Rightarrow k=\pm 6$

Take k = 6

$A^{2}=\begin{bmatrix} 36 & -32\\ 0 & 4 \end{bmatrix}$

Option 1)

$\begin{bmatrix} 4 &-32 \\ 0 & 36 \end{bmatrix}$

Option 2)

$\begin{bmatrix} 36 &0 \\ -32 & 4 \end{bmatrix}$

Option 3)

$\begin{bmatrix} 4 &0 \\ -32 & 36 \end{bmatrix}$

Option 4)

$\begin{bmatrix} 36 &-32 \\ 0& 4 \end{bmatrix}$

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