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Q8.     If A = \begin{bmatrix} 3 &1 \\ -1 & 2 \end{bmatrix}, show that A^2 -5A + 7I= 0.

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A = \begin{bmatrix} 3 &1 \\ -1 & 2 \end{bmatrix}

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

A^{2} = \begin{bmatrix} 9-1 &3+2 \\ -3-2 & -1+4 \end{bmatrix}

A^{2} = \begin{bmatrix} 8 &5 \\ -5 & 3 \end{bmatrix}

I= \begin{bmatrix} 1 &0 \\ 0& 1 \end{bmatrix}

To prove:  A^2 -5A + 7I= 0

L.H.S : A^2 -5A + 7I

= \begin{bmatrix} 8 &5 \\ -5 & 3 \end{bmatrix}-5 \begin{bmatrix} 3 &1 \\ -1 & 2 \end{bmatrix}+ 7 \begin{bmatrix} 1 &0 \\ 0& 1 \end{bmatrix}

=\begin{bmatrix} 8-15+7 &5-5+0 \\ -5+5+0& 3-10+7 \end{bmatrix}

 =\begin{bmatrix} 0 &0 \\ 0& 0 \end{bmatrix} =0=R.H.S

Hence, we proved that

  A^2 -5A + 7I= 0.

 

 

 

 

 

Posted by

seema garhwal

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