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  • Please solve RD Sharma class 12 chapter Algebra of matrices exercise 4.3 question 35maths textbook solution

Please solve RD Sharma class 12 chapter Algebra of matrices exercise 4.3 question 35 maths textbook solution

Answers (1)

Answer: k=1

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}{ll} 3 & -2 \\ 4 & -2 \end{array}\right] and A^{2}=k A-2 I_{2}

I_ 2 is an identity matrix of size 2. So, 2 I_{2}=2\left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right]=\left[\begin{array}{ll} 2 & 0 \\ 0 & 2 \end{array}\right]

Also given,

A^{2}=k A-2 I_{2}

Now, we will find the matrix for A^2, we get

A^{2}=A A=\left[\begin{array}{ll}3 & -2 \\ 4 & -2\end{array}\right]\left[\begin{array}{ll}3 & -2 \\ 4 & -2\end{array}\right] \\\\ A^{2}=\left[\begin{array}{cc}9-8 & -6+4 \\ 12-8 & -8+4\end{array}\right] \\\\\ A^{2}=\left[\begin{array}{ll}1 & -2 \\ 4 & -4\end{array}\right]----i

Now, we will find the value for kA, we get

k A=k\left[\begin{array}{ll}3 & -2 \\ 4 & -2\end{array}\right] \\\\ k A=\left[\begin{array}{ll}k \times 3 & k \times(-2) \\ k \times 4 & k \times(-2)\end{array}\right]

So, substituting corresponding values from equation i & ii in
A^{2}=k A-2 I_{2}

we get\left[\begin{array}{ll}1 & -2 \\ 4 & -4\end{array}\right]=\left[\begin{array}{ll}3 k & -2 k \\ 4 k & -2 k\end{array}\right]-\left[\begin{array}{ll}2 & 0 \\ 0 & 2\end{array}\right] \\\\\ \\ \left[\begin{array}{ll}1 & -2 \\ 4 & -4\end{array}\right]=\left[\begin{array}{ll}3 k-2 & -2 k-0 \\ 4 k-0 & -2 k-2\end{array}\right]

And to satisfy the above condition of equality, the corresponding entries of the matrices should be equal.Hence,

\begin{array}{l} 3 k=1+2 \\\\ 3 k=3 \\\\ k=\frac{3}{3}=1 \end{array}

Therefore, the value of k is 1

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