Solve the following system of linear equations by matrix inversion method: 2x + 3y – z = 9, x + y + z = 9, 3x – y – z = – 1 - Mathematics

Question

Solve the following system of linear equations by matrix inversion method:

2x + 3y – z = 9, x + y + z = 9, 3x – y – z = – 1

Sum

Solution

`[(2, 3, -1),(1, 1, 1),(3, -1, -1)][(x),(y),(z)] = [(9),(9),(-1)]`

AX = B

X = `"A"^-1"B"`

A = `[(2, 3, -1),(1, 1, 1),(3, -1, -1)]`

|A| = 2(–1+1) –3(–1 – 3) –1(–1 – 3)

= 0 + 12 + 4

=16 ≠ 0 A^-1 exists.

adj A = `[((-1 + 1), -(-1 - 3), (-1 - 3)),(-(-3 - 1),(-2 + 3), -(- 2 - 9)),((3 + 1), -(2 + 1), (2 - 3))]^"T"`

= `[(0, 4, -4),(4, 1, 11),(4, -3, -1)]^"T"`

= `[(0, 4, 4),(4, 1, -3),(-4, 11, -1)]`

`"A"^-1 = 1/|"A"|`

adj A = `1/16 [(0, 4, 4),(4, 1, -3),(-4, 11, -1)]`

X = `"A"^-1"B"`

`[(x),(y),(z)] = 1/16[(0, 4, 4),(4, 1, -3),(-4, 11, 1)][(9),(9),(-1)]`

= `1/16 [(0 + 36 - 4),(36 + 9 + 3),(- 36 + 99 + 1)]`

`[(x),(y),(z)] = 1/16 [(32), (48), (64)]`

= `[(2),(3),(4)]`

∴ x = 2, y = 3, z = 4

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Applications of Matrices: Solving System of Linear Equations

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Q 1. (iii)Q 1. (ii)Q 1. (iv)

Chapter 1: Applications of Matrices and Determinants - Exercise 1.3 [Page 33]

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Samacheer Kalvi Mathematics - Volume 1 and 2 [English] Class 12 TN Board

Chapter 1 Applications of Matrices and Determinants
Exercise 1.3 | Q 1. (iii) | Page 33

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