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

Chapter: 12th Mathematics : UNIT 1 : Applications of Matrices and Determinants

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 ,      … (1)

where A is a square matrix and non-singular. Since A is non-singular, A1 exists and A1 A = AA1 = I. Pre-multiplying both sides of (1) by A1, we get A1 ( AX ) = A1B. That is, ( A1 A) X = A1B. Hence, we get X = A1B.

 

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 ≠ 0. So, A−1  exists and A−1

Then, applying the formula X = A−1B , we get


So the solution is (x = −1, y = 4).

 

Example 1.23

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

2x1 + 3x2 + 3x3 = 5,

x1 – 2x2 + x3 = -4,

3x1 – x2 – 2x3 = 3

Solution

The matrix form of the system is AX = B,where


So, the solution is ( x1 = 1, x2 = 2, x3 = −1) .

 

Example 1.24

If , find the products AB and BA and hence solve the system of equations x − y + z = 4, x – 2y – 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 = −1).

 

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

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12th Mathematics : UNIT 1 : Applications of Matrices and Determinants


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