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  • Explain solution for rd sharma class class 12 chapter Algebra of matrices exercise 4.3 question 38 math

Explain solution for rd sharma class class 12 chapter Algebra of matrices exercise 4.3 question 38 math

Answers (1)

Answer:  \lambda=4 and  \mu=-1

Hint: Iis 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}2 & 3 \\ 1 & 2\end{array}\right] andI=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]

\\\\ A^{2}=\lambda A+\mu I

So,

\mu I=\mu\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]=\left[\begin{array}{ll}\mu & 0 \\ 0 & \mu\end{array}\right]

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

A^{2}=A A=\left[\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right]\left[\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right] \\\\ \\ A^{2}=\left[\begin{array}{ll}4+3 & 6+6 \\ 2+2 & 3+4\end{array}\right] \\\\ A^{2}=\left[\begin{array}{cc}7 & 12 \\ 4 & 7\end{array}\right] ....(1)

Now, we will find the matrix for , \lambda Awe get

\lambda A=\lambda\left[\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right] \\\\ \lambda A=\left[\begin{array}{ll}2 \lambda & 3 \lambda \\ 1 \lambda & 2 \lambda\end{array}\right] ....(ii)

But given, A^{2}=\lambda A+\mu I

So, substituting corresponding values from equation i & ii we get

\left[\begin{array}{cc}7 & 12 \\ 4 & 7\end{array}\right]=\left[\begin{array}{ll}2 \lambda & 3 \lambda \\ 1 \lambda & 2 \lambda\end{array}\right]+\left[\begin{array}{ll}\mu & 0 \\ 0 & \mu\end{array}\right] \\\\\\ \left[\begin{array}{cc}7 & 12 \\ 4 & 7\end{array}\right]=\left[\begin{array}{ll}2 \lambda+\mu & 3 \lambda+0 \\ 1 \lambda+0 & 2 \lambda+\mu\end{array}\right]

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

\lambda+0=4 \\\\ \lambda=4\\\\ 2 \lambda+\mu=7\\\\ 8+\mu=7\\\\ \mu=7-8=-1

Therefore, the value of \lambda = 4 and \mu = -1

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