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# Matrix Inversion Method

This method can be applied only when the coefficient matrix is a square matrix and non-singular.

Matrix Inversion Method

This method can be applied only when the coefficient matrix is a square matrix and non-singular.

Consider the matrix equation

AX   =  B ,      ŌĆ” (1)

where A is a square matrix and non-singular. Since A is non-singular, AŌłÆ1 exists and AŌłÆ1 A = AAŌłÆ1 = I. Pre-multiplying both sides of (1) by AŌłÆ1, we get AŌłÆ1 ( AX ) = AŌłÆ1B. That is, ( AŌłÆ1 A) X = AŌłÆ1B. Hence, we get X = AŌłÆ1B.

### Example 1.22

Solve the following system of linear equations, using matrix inversion method:

5x + 2 y = 3, 3x + 2 y = 5 .

### Solution

The matrix form of the system is AX = B , where We find |A| = = 10 - 6= 4 ŌēĀ 0. So, AŌłÆ1  exists and AŌłÆ1 Then, applying the formula X = AŌłÆ1B , we get So the solution is (x = ŌłÆ1, y = 4).

### Example 1.23

Solve the following system of equations, using matrix inversion method:

2x1 + 3x2 + 3x3 = 5,

x1 ŌĆō 2x2 + x3 = -4,

3x1 ŌĆō x2 ŌĆō 2x3 = 3

### Solution

The matrix form of the system is AX = B,where So, the solution is ( x1 = 1, x2 = 2, x3 = ŌłÆ1) .

Example 1.24

If , find the products AB and BA and hence solve the system of equations x ŌłÆ y + z = 4, x ŌĆō 2y ŌĆō 2z = 9, 2x + y +3z =1.

Solution Writing the given system of equations in matrix form, we get Hence, the solution is (x = 3, y = - 2,  z = ŌłÆ1).

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12th Mathematics : UNIT 1 : Applications of Matrices and Determinants : Matrix Inversion Method | Definition, Formulas, Solved Example Problems