Solve the following system of equations by using Matrix inversion method. 2x – y + 3z = 9, x + y + z = 6, x

Engineering

Mathematics

Multiplication of Matrices

Symmetric and Skew Symmetric Matrices

Question

Solve the following system of equations by using Matrix inversion method.
2x – y + 3z = 9, x + y + z = 6, x – y + z = 2

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Given

2x – y + 3z = 9 ………..(1)

x + y + z = 6 …………..(2)

x 0 y + z = 2 …………….(2)

using matrix inversion method

AX = B

[\begin{array}{ccc} 2 & - 1 & 3 \\ 1 & 1 & 1 \\ 1 & - 1 & 1 \end{array}] [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 9 \\ 6 \\ 2 \end{array}]

and X = A^–1B where

A^{- 1} = \frac{adj (A)}{∣ A ∣}

Now

adjA = [\begin{array}{ccc} 2 & - 2 & - 4 \\ 0 & - 1 & 1 \\ - 2 & - 1 & 3 \end{array}]

|A| = 2(1 + 1) – 1(1 – 1) + 3(–1 – 1)

|A| = 4 – 6

|A| = – 2.

Now

A^{- 1} = \frac{- 1}{2} [\begin{array}{ccc} 2 & - 2 & - 4 \\ 0 & - 1 & 1 \\ - 2 & - 1 & 3 \end{array}]

Now X = A^–1B

[\begin{array}{c} x \\ y \\ z \end{array}] = \frac{- 1}{2} [\begin{array}{ccc} 2 & - 2 & - 4 \\ 0 & - 1 & 1 \\ - 2 & - 1 & 3 \end{array}] [\begin{array}{c} 9 \\ 6 \\ 2 \end{array}]

[\begin{array}{c} x \\ y \\ z \end{array}] = \frac{- 1}{2} [\begin{array}{c} 18 - 12 - 8 \\ 0 - 6 + 2 \\ - 18 - 6 + 6 \end{array}] = \frac{- 1}{2} [\begin{array}{c} - 2 \\ - 4 \\ - 18 \end{array}]

[\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 1 \\ 2 \\ 9 \end{array}]

Hence [x = 1, y = 2 & z = 9.

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