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Solution of Simultaneous Equations with Matrices

⁄
AA 1 – = 4 3 1 – 3 2 = 4 – 3 – 6 + 6 = 1 0 = I
2 2 – 1 2 2 – 2 – 3 + 4 0 1

4.11 Solution of Simultaneous Equations with Matrices

Consider the relation
AX = B (4.48)

where and are matrices whose elements are known, and is a matrix (a column vector)
X
B
A
whose elements are the unknowns. We assume that and are conformable for multiplication.
A
X
Multiplication of both sides of (4.48) by A 1 – yields:
1 –
1 –
1 –
A AX = A B = IX = A B (4.49)
or
1 –
X=A B (4.50)
Therefore, we can use (4.50) to solve any set of simultaneous equations that have solutions. We
will refer to this method as the inverse matrix method of solution of simultaneous equations.

Example 4.16

Given the system of equations

⎧ 2x + 3x + x = 9 ⎫
2
1
3
⎪ ⎪
⎨ x + 2x + 3x = 6 ⎬ (4.51)
1
3
2
⎪ ⎪
⎩ 3x + x + 2x = 8 ⎭
3
2
1
,
,
compute the unknowns x x and x 3 using the inverse matrix method.
1
2
Solution:
In matrix form, the given set of equations is AX = B where
2 3 1 x 1 9
A = 1 2 3 , X = x , 2 B = 6 (4.52)
3 1 2 x 3 8
Then,

Numerical Analysis Using MATLAB® and Excel®, Third Edition 4−25
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