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  • Provide solution for rd sharma math class class 12 chapter Algebra of matrices exercise 4.3 question 36

Provide solution for rd sharma math class class 12 chapter Algebra of matrices exercise 4.3 question 36

Answers (1)

Answer: k=7

Hint: 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}{cc}1 & 0 \\ -1 & 7\end{array}\right] and A^{2}-8 A+k l=0

I is an identity matrix. So, k I=k\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]=\left[\begin{array}{ll}k & 0 \\ 0 & k\end{array}\right]

Now, we have to find A^2, we get

A^{2}=A A=\left[\begin{array}{cc}1 & 0 \\ -1 & 7\end{array}\right]\left[\begin{array}{cc}1 & 0 \\ -1 & 7\end{array}\right] \\\\ \\ A^{2}=\left[\begin{array}{cc}1+0 & 0+0 \\ -1-7 & 0+49\end{array}\right] \\\\ \\ A^{2}=\left[\begin{array}{cc}1 & 0 \\ -8 & 49\end{array}\right] ...(i)

Now, we will find the matrix 8A, we get

8 A=8\left[\begin{array}{cc}1 & 0 \\ -1 & 7\end{array}\right]=\left[\begin{array}{cc}8 \times 1 & 8 \times 0 \\ 8 \times(-1) & 8 \times 7\end{array}\right] \\\\ 8 \boldsymbol{A}=\left[\begin{array}{cc}8 & 0 \\ -8 & 56\end{array}\right] ... (ii)

   So, substituting corresponding values from equation i & ii in equation


A^{2}-8 A+k I=0 \\\\ \Rightarrow\left[\begin{array}{cc}1 & 0 \\ -8 & 49\end{array}\right]-\left[\begin{array}{cc}8 & 0 \\ -8 & 56\end{array}\right]+\left[\begin{array}{cc}k & 0 \\ 0 & k\end{array}\right]=0 \\\\\\ \Rightarrow\left[\begin{array}{cc}1-8+k & 0-0+0 \\ -8+8+0 & 49-56+k\end{array}\right]=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]

And to satisfy the above conditions of equality, the corresponding entries of the matrices should be equal

Hence,

1-8+k-0

k=8-1=7

Therefore, the value of k=7

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