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  • Need solution for RD Sharma maths class 12 chapter Algebra of matrices exercise 4.3 question 29 math

Need solution for RD Sharma maths class 12 chapter Algebra of matrices exercise 4.3 question 29 math

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Answer: Hence, proved A^{2}-5 A+7 I_{2}=0

Hint: I_2 is the identity matrix of size 2.         

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

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

Prove:A^{2}-5 A+7 I_{2}=0

Consider:
A^{2}-5 A+7 I_{2} \\\\ =\left[\begin{array}{cc} 3 & 1 \\ -1 & 2 \end{array}\right]\left[\begin{array}{cc} 3 & 1 \\ -1 & 2 \end{array}\right]-5\left[\begin{array}{cc} 3 & 1 \\ -1 & 2 \end{array}\right]+7\left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right] \\\\\

=\left[\begin{array}{cc} 8-15+7 & 5-5+0 \\ -5+5+0 \quad 3-10+7 \end{array}\right] \\\\\\ =\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right]

Hence, A^{2}-5 A+7 I_{2}=0

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