Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • School
  • If <img alt="A =\begin{bmatrix} 1 &2 &-3 \\ 3 &2 &-2 \\ 2 &-1 &1 \end{bmatrix}" src="https://learn.careers360.com/latex-image/?A%20%3D%5Cbegin%7Bbmatrix%7D%201%20%262%20%26-3%20%5C%5C%203%20%262%20%26-2%20%5C%5C%202%20%26-1%20%2

If A =\begin{bmatrix} 1 &2 &-3 \\ 3 &2 &-2 \\ 2 &-1 &1 \end{bmatrix} , then find A^{-1} and use it to solve the following system of the equations :
x + 2y -3z = 6
3x + 2y - 2z = 3
2x - y + z = 2

 

 
 
 
 
 

Answers (1)

Given:

A =\begin{bmatrix} 1 &2 &-3 \\ 3 &2 &-2 \\ 2 &-1 &1 \end{bmatrix}

\\|A| = 1(2-2)-2(3+4)-3(-3-4) \\ |A|= -14+21 =7 \\

\\ \operatorname{Adj}(A)=\left[\begin{array}{rrr} 0 & -7 & -7 \\ 1 & 7 & 5 \\ 2 & -7 & -4 \end{array}\right]^T

\\ \operatorname{Adj}(A)=\left[\begin{array}{rrr} 0 & 1 & 2 \\ -7 & 7 & -7 \\ -7 & 5 & -4 \end{array}\right] \\\\ A^{-1}=\frac{1}{|A|} \text { Adj } A \\\\ =\frac{1}{7}\left[\begin{array}{rrr} 0 & 1 & 2 \\ -7 & 7 & -7 \\ -7 & 5 & -4 \end{array}\right] \\\\ =\left[\begin{array}{rrr} 0 & \frac{1}{7} & \frac{2}{7} \\ -1 & 1 & -1 \\ -1 & \frac{5}{7} & -\frac{4}{7} \end{array}\right]

x + 2y -3z = 6
3x + 2y - 2z = 3
2x - y + z = 2

The given system of linear equations can be written in matirx form as AX = B

\\\mathrm{X}=\left[\begin{array}{l} \mathrm{x} \\ \mathrm{y} \\ \mathrm{z} \end{array}\right] \\\\ \mathrm{B}=\left[\begin{array}{l} 6 \\ 3 \\ 2 \end{array}\right]\\

X=A^{-1} B

\\ \left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{rrr} 0 & \frac{1}{7} & \frac{2}{7} \\ -1 & 1 & -1 \\ -1 & \frac{5}{7} & -\frac{4}{7} \end{array}\right]\left[\begin{array}{l} 6 \\ 3 \\ 2 \end{array}\right] \\ \\ \\ \left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{r} 1 \\ -5 \\ -5 \end{array}\right]

Hence x = 1,  y=-5, z=-5
 

Posted by

Safeer PP

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

CBSE asks schools to ensure error-free LOC forms for Classes 10, 12; edit window from October 13
anam-khan

‘You ruined my career’: Students hit back at CBSE over 'abrupt' removal of additional subjects
anam-khan

‘Careers are at stake’: CBSE removes additional subject option for private students without prior notice

Latest Question