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If A=\begin{bmatrix} -3 &2 \\ 1& -1 \end{bmatrix}and I=\begin{bmatrix} 1 &0 \\ 0& 1 \end{bmatrix} , find scalar k so that A^{2} + I = kA.

 

 
 
 
 
 

Answers (1)

Given: 

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

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

A^{2}=AA = \left[\begin{array}{cc} -3 & 2 \\ 1 & -1 \end{array}\right] \left[\begin{array}{cc} -3 & 2 \\ 1 & -1 \end{array}\right]

A^{2}=\left[\begin{array}{cc} 11 & -8 \\ -4 & 3 \end{array}\right]

A^{2} + I = kA

\left[\begin{array}{cc} 11 & -8 \\ -4 & 3 \end{array}\right] + \left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right] = k\left[\begin{array}{cc} -3 & 2 \\ 1 & -1 \end{array}\right]

\left[\begin{array}{cc} 12 & -8 \\ -4 & 4 \end{array}\right] =\left[\begin{array}{cc} -3k & 2k \\ k & -k \end{array}\right]

k = -4

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Safeer PP

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