Page 281 - Advanced engineering mathematics

8.5 Cramer’s Rule 261

THEOREM 8.5 Cramer’s Rule

Let A be a nonsingular n × n matrix of numbers, and B be an n × 1 matrix of numbers. Then the
unique solution of AX = B is determined by
1
x k = |A(k;B)| (8.7)
|A|
for k = 1,2,··· ,n, where A(k;B) is the matrix obtained from A by replacing column k of A
with B.

It is easy to see why this works. Let
⎛ ⎞
b 1
b 2
⎜ ⎟
B = ⎜ . ⎟.
⎜ ⎟
.
⎝ . ⎠
b n
Multiply column k of A by x k . This multiplies the determinant of A by x k :

a 11 a 12 a 1k x k a 1n
··· ···

a 21 a 22 a 2k x k a 2n
··· ···

x k |A|= . . . . . . .
. . . . . . . . . . . .

a n1 a n2 ··· a nk x k ··· a nn
For each j = k add x j times column j to column k in the last determinant. Since this operation
does not change the value of a determinant, then

··· ···
a 11 a 12 a 11 x 1 + ··· + a 1n x n a 1n

··· ···
a 21 a 22 a 21 x 1 + ··· + a 2n x n a 2n

x k |A|= . . . . .

. . . . . . . . . . .
.
.

a n1 a n2 ··· a n1 x 1 + ··· + a nn x n ··· a nn

a 11 a 12 b 1 a 1n
··· ···

a 21 a 22 b 2 a 2n
··· ···
= . . . . . . =|A(k;B)|

. . . . . . . . . . . .

a n1 a n2 ··· b n ··· a nn
and this gives us equation (8.7).
EXAMPLE 8.7
Solve the system
x 1 − 3x 2 − 4x 3 = 1
−x 1 + x 2 − 3x 3 = 14
x 2 − 3x 3 = 5.
The matrix of coefficients is
⎛ ⎞
1 −3 −4
A = −1 1 −3 ⎠ .
⎝
0 1 −3
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October 14, 2010 14:26 THM/NEIL Page-261 27410_08_ch08_p247-266