**MATHS :: Lecture 11 :: ****
INVERSE OF A MATRIX
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**INVERSE OF A MATRIX**

**Definition**

Let A be any square matrix. If there exists another square matrix B Such that AB = BA = I (I is a unit matrix) then B is called the inverse of the matrix A and is denoted by A-1.

The cofactor method is used to find the inverse of a matrix. Using matrices, the solutions of simultaneous equations are found.** **

Working Rule to find the inverse of the matrix

Step 1: Find the determinant of the matrix.

Step 2: If the value of the determinant is non zero proceed to find the inverse of the matrix.

Step 3: Find the cofactor of each element and form the cofactor matrix.

Step 4: The transpose of the cofactor matrix is the adjoint matrix.

Step 5: The inverse of the matrix A-1 =

Finding Matrix Inverse

Cramer's Rule, Inverse Matrix, and Volume

Find the inverse of the matrix

Solution

Let A =

Step 1

Step 2

The value of the determinant is non zero

\A-1 exists.

Step 3

Let Aij denote the cofactor of

*aij*in

Step 4

The matrix formed by cofactors of element of determinant is

\adj A =

Step 5

=

SOLUTION OF LINEAR EQUATIONS

Let us consider a system of linear equations with three unknowns

The matrix form of the equation is AX=B where

**is a 3x3 matrix**

X = and B =

Here AX = B

Pre multiplying both sides by A‑1.

(A-1 A)X= A-1B

We know that A-1 A= A A-1=I

\ I X= A-1B

since IX = X

Hence the solution is X = A-1B.

Example

Solve the

*x + y + z = 1,*3

*x +*5

*y +*6

*z = 4,*9

*x +*26

*y +*36

*z =16*by matrix method.

Solution

The given equations are

*x + y + z = 1,*

3

*x +*5

*y +*6

*z = 4,*

9

*x +*26

*y +*36

*z =16*

Let A= , X=, B=

The given system of equations can be put in the form of the matrix equation AX=B

The value of the determinant is non zero

\ A-1 exists.

Let Aij (i, j = 1,2,3) denote the cofactor of *aij *in

The matrix formed by cofactors of element of determinant is

\adj A =

We Know that X=A-1B

\ =

=

*x*** = 0, y = 2, z = -1. **

**SOLUTION BY DETERMINANT (CRAMER'S RULE)**

Let the equations be

……………………. (1)

Consider the determinant

When D ≠ 0, the unique solution is given by

**Example**

Solve the equations

*x +*2

*y +*5

*z =23,*3

*x + y +*4

*z = 26,*6

*x + y +*7

*z = 47*by determinant method (Cramer’s Rule).

**Solution**

The equations are

*x + 2y +*5

*z*=23,

3

*x + y +*4

*z*= 26,

6

*x + y +*7

*z*= 47

By Cramer’s rule

Þ

*x*= 4,

*y*= 2,

*z*= 3.

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