Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • School
  • Explain solution for rd sharma class class 12 chapter Algebra of matrices exercise 4.3 question 34 math

Explain solution for rd sharma class class 12 chapter Algebra of matrices exercise 4.3 question 34 math

Answers (1)

Answer:

A^{2}-5 A-7 I=0, A^{4}=\left[\begin{array}{cc}39 & 55 \\ -55 & -16\end{array}\right]

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}3 & 1 \\ -1 & 2\end{array}\right]

Prove:A^{2}-5 A+7 I=0

Solution: I is identity matrix so

7 I=7\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]=\left[\begin{array}{ll}7 & 0 \\ 0 & 7\end{array}\right]

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

A^{2}=A A=\left[\begin{array}{cc}3 & 1 \\ -1 & 2\end{array}\right]\left[\begin{array}{cc}3 & 1 \\ -1 & 2\end{array}\right]\\\\ A^{2}=\left[\begin{array}{cc}9-1 & 3+2 \\ -3-2 & -1+4\end{array}\right]\\\\ A^{2}=\left[\begin{array}{cc}8 & 5 \\ -5 & 3\end{array}\right] ...(i)

Now, we will find the matrix for  5A, we get

5 A=5\left[\begin{array}{cc}3 & 1 \\ -1 & 2\end{array}\right]\\\\ 5 A=\left[\begin{array}{cc}15 & 5 \\ -5 & 10\end{array}\right] ... (ii)

So, substituting corresponding values from equation  i &  ii in
A^{2}-5 A+7 I

we get

\begin{array}{l} =\left[\begin{array}{cc} 8 & 5 \\ -5 & 3 \end{array}\right]-\left[\begin{array}{cc} 15 & 5 \\ -5 & 10 \end{array}\right]+\left[\begin{array}{ll} 7 & 0 \\ 0 & 7 \end{array}\right] \\\\ =\left[\begin{array}{cc} 8-15+7 & 5-5+0 \\ -5+5+0 & 3-10+7 \end{array}\right] \\\\ =\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right]=0 \end{array}

\therefore A^{2}-5 A+7 I=0

Hence, proved.

We will find A^4 

A^{2}-5 A+7 I=0

Multiply both sides by A^2, we get

A^{2}\left(A^{2}-5 A+7 I\right)=A^{2}(0) \\\\ A^{4}-5 A^{2} A+7 I A^{2}=0 \\\\ A^{4}=5 A^{2} A-7 I A^{2}\\\\ A^{4}=5 A^{2} A-7 A^{2}

Now we will substitute the corresponding values we get

A^{4}=5\left[\begin{array}{cc}8 & 5 \\ -5 & 3\end{array}\right]\left[\begin{array}{cc}3 & 1 \\ -1 & 2\end{array}\right]-7\left[\begin{array}{cc}8 & 5 \\ -5 & 3\end{array}\right]\\\\ \\ A^{4}=5\left[\begin{array}{cc}24-5 & 8+10 \\ -15-3 & -5+6\end{array}\right]-7\left[\begin{array}{cc}8 & 5 \\ -5 & 3\end{array}\right] \\\\\\ A^{4}=5\left[\begin{array}{cc}19 & 18 \\ -18 & 1\end{array}\right]-7\left[\begin{array}{cc}8 & 5 \\ -5 & 3\end{array}\right]\\\\\ A^{4}=\left[\begin{array}{cc}5 \times 19 & 5 \times 18 \\ -18 \times 5 & 1 \times 5\end{array}\right]-\left[\begin{array}{cc}7 \times 8 & 7 \times 5 \\ -5 \times 7 & 3 \times 7\end{array}\right] \\\\\\ A^{4}=\left[\begin{array}{cc}95 & 90 \\ -90 & 5\end{array}\right]-\left[\begin{array}{cc}56 & 35 \\ -35 & 21\end{array}\right]\\\\\\ A^{4}=\left[\begin{array}{cc}95-56 & 90-35 \\ -90+35 & 5-21\end{array}\right] \\\\\\A^{4}=\left[\begin{array}{cc}39 & 55 \\ -55 & -16\end{array}\right]

Posted by

infoexpert22

View full answer

Crack CUET with india's "Best Teachers"

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

Similar Questions

Trending Articles/News

anam-khan

Board Exam Result 2025 Live: UK Board 10th, 12th results tomorrow; AP SSC, TS Inter, UP, CBSE updates
anam-khan

When will CBSE Class 12 results 2025 be declared? Last two years' trends
anam-khan

CBSE Class 10, 12 board exams 2025 from April 7 to 11 for sports students

Latest Question