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Provide solution for rd sharma math class 12 chapter Algebra of matrices exercise 4.3 question 55

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Answer:

A^{2}-5 A+4 I=\left[\begin{array}{ccc} -1 & -1 & -3 \\ -1 & -3 & -10 \\ -5 & 4 & 2 \end{array}\right] \text { and } X=\left[\begin{array}{ccc} 1 & 1 & 3 \\ 1 & 3 & 10 \\ 5 & -4 & -2 \end{array}\right]

Hint: I is an identity matrix,

I_{2}=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \text { and } I_{3}=\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]

Given:

A=\left[\begin{array}{ccc} 2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 0 \end{array}\right]

We have to find  A^{2}-5 A+4 I

\\\\ =\left[\begin{array}{ccc} 2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 0 \end{array}\right]\left[\begin{array}{ccc} 2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 0 \end{array}\right]-5\left[\begin{array}{ccc} 2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 0 \end{array}\right]+4\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\\\\\\ =\left[\begin{array}{ccc} 4+0+1 & 0+0-1 & 2+0+0 \\ 4+2+3 & 0+1-3 & 2+3+0 \\ 2-2+0 & 0-1-0 & 1-3+0 \end{array}\right]-\left[\begin{array}{ccc} 10 & 0 & 5 \\ 10 & 5 & 15 \\ 5 & -5 & 0 \end{array}\right]+\left[\begin{array}{ccc} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{array}\right]\\\\\\ =\left[\begin{array}{ccc} 5 & -1 & 2 \\ 9 & -2 & 5 \\ 0 & -1 & -2 \end{array}\right]-\left[\begin{array}{ccc} 10 & 0 & 5 \\ 10 & 5 & 15 \\ 5 & -5 & 0 \end{array}\right]+\left[\begin{array}{ccc} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{array}\right]

\\\\=\left[\begin{array}{ccc} 5-10+4 & -1-0+0 & 2-5+0 \\ 9-10+0 & -2-5+4 & 5-15+0 \\ 0-5+0 & -1+5+0 & -2-0+4 \end{array}\right]\\\\\\ =\left[\begin{array}{ccc} -1 & -1 & -3 \\ -1 & -3 & -10 \\ -5 & 4 & 2 \end{array}\right]\\\\\\ A^{2}-5 A+4 I=\left[\begin{array}{ccc} -1 & -1 & -3 \\ -1 & -3 & -10 \\ -5 & 4 & 2 \end{array}\right]

Now given

A^{2}-5 A+4 I+X=0

 

\\X=-\left(A^{2}-5 A+4 I\right) \\\\\\ X=(-)\left[\begin{array}{ccc} -1 & -1 & -3 \\ -1 & -3 & -10 \\ -5 & 4 & 2 \end{array}\right] \\\\\\ X=\left[\begin{array}{ccc} 1 & 1 & 3 \\ 1 & 3 & 10 \\ 5 & -4 & -2 \end{array}\right]

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