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Please help! - Matrices and Determinants - JEE Main-2

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}

 
Answers (2)
97 Views
N neha

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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