Chapter 23
The solution of simultaneous
equations by matrices and
determinants
23.1 Solution of simultaneous
equations by matrices
(a) The procedure for solving linear simultaneous
equations in two unknowns using matrices is:
(i) write the equations in the form
a 1 x + b 1 y = c 1
a 2 x + b 2 y = c 2
(ii) write the matrix equation corresponding to
these equations,
i.e.
a 1 b 1
a 2 b 2
×
x
y
=
c 1
c 2
(iii) determine the inverse matrix of
a 1 b 1
a 2 b 2
i.e.
1
a 1 b 2 − b 1 a 2
b 2 −b 1
−a 2
a 1
(from Chapter 22)
(iv) multiply each side of (ii) by the inverse
matrix, and
(v) solve for x and y by equating corresponding
elements.
Problem 1. Use matrices to solve the
simultaneous equations:
3x + 5y − 7 = 0
( 1 )
4x − 3y − 19 = 0
( 2 )
(i) Writing the equations in the a 1 x + b 1 y = c form
gives:
3x + 5y = 7
4x − 3y = 19
(ii) The matrix equation is
3
5
4 −3
×
x
y
=
7
19
(iii) The inverse of matrix
3
5
4 −3
is
1
3 × (−3) − 5 × 4
−3 −5
−4
3
i.e.
⎛
⎜
⎝
3
29
5
29
4
29
−3
29
⎞
⎟
⎠
(iv) Multiplying each side of (ii) by (iii) and
remembering that A × A −1 = I , the unit matrix,
gives:
1 0
0 1
x
y
=
⎛
⎜
⎜
⎝
3
29
5
29
4
29
−3
29
⎞
⎟
⎟
⎠ ×
7
19