http://www.ck12.org Chapter 8. Systems and Matrices
[a b
c d][x
y]
=
[e
f]
x=∣∣
∣∣e b
f d∣∣
∣∣
∣∣
∣∣a b
c d∣∣
∣∣
y=∣∣
∣∣a e
c f∣∣
∣∣
∣∣
∣∣a b
c d∣∣
∣∣
This is a fantastic improvement over solving systems using substitution or elimination. Cramer’s Rule also works
with larger order matrices. For a system of 3 variables and 3 equations the reasoning is identical.
ax+by+cz=j
dx+ey+f z=k
gx+hy+iz=lThe system can be represented as a matrix.
a b c
d e f
g h i
·
x
y
z
=
j
k
l
The three solutions can be represented as a ratio of determinants.
x=∣∣
∣∣
∣∣
j b c
k e f
l h i∣∣
∣∣
∣∣
∣∣
∣∣
∣∣
a b c
d e f
g h i∣∣
∣∣
∣∣
y=∣∣
∣∣
∣∣
a j c
d k f
g l i∣∣
∣∣
∣∣
∣∣
∣∣
∣∣
a b c
d e f
g h i∣∣
∣∣
∣∣
z=∣∣
∣∣
∣∣
a b j
d e k
g h l∣∣
∣∣
∣∣
∣∣
∣∣
∣∣
a b c
d e f
g h i∣∣
∣∣
∣∣
Remember that evaluating the determinants of 3×3 matrices using Sarrus’s rule is very efficient.