If A= \begin{bmatrix} 2 & -3 &5 \\ 3& 2 &-4 \\ 1&1 & -2 \end{bmatrix} then find A^{-1}.hence solve the following system of equation:
2x-3y+5z= 11,3x+2y-4z= -5,x+y-2z= -3.

 

 

 

 
 
 
 
 

Answers (1)

given= A= \begin{bmatrix} 2 & -3 &5 \\ 3& 2 &-4 \\ 1&1 & -2 \end{bmatrix}
\left | A \right |= 2\left ( -1+1 \right )+3\left ( -6+4 \right )+5\left ( 3-2 \right )
\left | A \right |=3\left ( -2 \right )+5\left ( 1 \right )
\left | A \right |=-1\neq 0
\therefore A^{-1} exists
cofactors of A,
a_{11}= 0,a_{21}= -1,a_{31}= 2
a_{12}= 2,a_{22}= -9,a_{32}= 23
a_{13}= 1,a_{23}= -5,a_{33}= 13
Adj A= \begin{bmatrix} 0 & 2 &1 \\ -1&-9 &-5 \\ 2 & 23 &13 \end{bmatrix}
Adj A= \begin{bmatrix} 0 & -1 &2 \\ 2&-9 &23 \\ 1 & -5 &13 \end{bmatrix}
A^{-1}= \frac{ Adj A}{\left | A \right |}
A^{-1}= \frac{1}{-1}\begin{bmatrix} 0 & -1 &2 \\ 2&-9 &23 \\ 1 & -5 &13 \end{bmatrix}=\begin{bmatrix} 0 & 1 &-2 \\ -2 &9 &-23 \\ -1 & 5 & -13 \end{bmatrix} --\left ( i \right )
now given equation are
2x-3y+5x= 11
3x+2y-A= -5x+y-22= -3
A= \begin{bmatrix} 2 & -3 & 5\\ 3& 2 & -1\\ 1 &1 & -2 \end{bmatrix},X= \begin{bmatrix} x\\y \\z \end{bmatrix},B= \begin{bmatrix} 11\\-5 \\-3 \end{bmatrix}
AX= B
\therefore X= A^{-1}B
\begin{bmatrix} x\\ y\\ z \end{bmatrix}= \begin{bmatrix} 0 & 1 &-2 \\ -2& 9 & -23\\ -1& 5 & -13 \end{bmatrix}\begin{bmatrix} 11\\ -5\\ -3 \end{bmatrix} from\left ( i \right )
\Rightarrow \begin{bmatrix} x\\y \\ z \end{bmatrix}= \begin{bmatrix} 0 -5 +6 \\ -22 -45 +69 \\ -11 -25 +39 \end{bmatrix}
\Rightarrow \begin{bmatrix} x\\y \\ z \end{bmatrix}= \begin{bmatrix} 1\\2 \\ 3 \end{bmatrix}
hence x= 1,y= 2,z= 3
 

Preparation Products

Knockout NEET Sept 2020

An exhaustive E-learning program for the complete preparation of NEET..

₹ 15999/- ₹ 6999/-
Buy Now
Rank Booster NEET 2020

This course will help student to be better prepared and study in the right direction for NEET..

₹ 9999/- ₹ 4999/-
Buy Now
Knockout JEE Main Sept 2020

An exhaustive E-learning program for the complete preparation of JEE Main..

₹ 12999/- ₹ 6999/-
Buy Now
Test Series NEET Sept 2020

Take chapter-wise, subject-wise and Complete syllabus mock tests and get in depth analysis of your test..

₹ 4999/- ₹ 2999/-
Buy Now
Knockout NEET May 2021

An exhaustive E-learning program for the complete preparation of NEET..

₹ 22999/- ₹ 11999/-
Buy Now
Exams
Articles
Questions