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  • Need solution for RD Sharma maths class 12 chapter Algebra of matrices exercise 4.3 question 37 math

Need solution for RD Sharma maths class 12 chapter Algebra of matrices exercise 4.3 question 37math

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Answer:  Hence provedf(A)=0

Hint: I is an identity matrix.

Matrix multiplication is only possible, when the number of columns of first matrix is equal to the number of rows of second matrix.

Given: A=\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right]and f(x)=x^{2}-2 x-3

To show that f(A)=0

Substitute x=A in f(x) we get

f(A)=A^{2}-2 A-3 I---i\\\\ f(A)=\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right]\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right]-2\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right]-3\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\\\\ =\left[\begin{array}{ll}1+4 & 2+2 \\ 2+2 & 4+1\end{array}\right]-\left[\begin{array}{ll}2 & 4 \\ 4 & 2\end{array}\right]-\left[\begin{array}{ll}3 & 0 \\ 0 & 3\end{array}\right] =\left[\begin{array}{ll}5 & 4 \\ 4 & 5\end{array}\right]-\left[\begin{array}{ll}2 & 4 \\ 4 & 2\end{array}\right]-\left[\begin{array}{ll}3 & 0 \\ 0 & 3\end{array}\right]\\\\ =\left[\begin{array}{lll}5-2-3 & 4-4-0 \\ 4-4-0 & 5-2-3\end{array}\right]\\\\ =\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]\\\\ =0\\\\

So, f (A) = 0

Hence, proved

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