Get Answers to all your Questions

header-bg qa

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

Posted by

infoexpert22

View full answer

Crack CUET with india's "Best Teachers"

  • HD Video Lectures
  • Unlimited Mock Tests
  • Faculty Support
cuet_ads