Determinants and Multiplicative Inverses of Matrices
A Square Matrix is a matrix that has the same number of rows and columns. Each square
matrix has a value associated with it known as the determinant. Finding the determinant of a
2 2 matrix is very simple. Higher dimensions matrices are much more complicated.
a b
Let matrix A
c d
5
Example 1 :
Example 2:
Example 3:
2
-1 3
3 7
2 1
6 2
3 1
the determinant of A = det A =
5(3) 2( 1)
c d
ad
bc
17
3(1) 7(2)
6(1) 2(3)
a b
11
0
As you can see determinants can have values that are positive, negative or zero. The
value of a determinant does have a special meaning later but determinants that have a value
of ZERO are very important!!
a1
The situation is more complicated for a 3 x 3 matrix. Let A = b1
c1
det A
a1
b1
c1
a2
b2
c2
a3
b3
c3
a1
b2
c2
b3
c3
a2
b1
c1
b3
c3
a3
b1
c1
a2
b2
c2
a3
b3
c3
b2
c2
This can be simplified as follows:
write down the original determinant and then write the first two columns down again:
a1
b1
c1
a2
b2
c2
a3
b3
c3
a1
b1
c1
a2
b2
c2
We now multiply the three terms shown by the lines. On the lines slanting down towards the
bottom right we ADD these terms together. On the lines slanting upward to the top right we
SUBTRACT these terms.
Example 4: Evaluate the following determinant:
1
2
4
-3
3
1
3
-2 5
determinant = 1(3)(5)+(2)(1)(3)+(4)(-3)(-2)
-(3)(3)(4)-(-2)(1)(1)-(5)(-3)(2) = 41
Example 5:
-4 -6
2
5
1
3
-2
4
-3
determinant = ( 4)(1)( 3) ( 6)(3)( 2) (2)(5)(4)
( 2)(1)(2) (4)(3)( 4) ( 3)(5)( 6)
50
Inverse of a Matrix
In the Real Number System, the multiplicative identity is the number 1. Here, x 1
1
x.
1
The multiplicative inverse is also called the reciprocal and is written x . Here, x x 1.
With matrices we have the same properties. The multiplicative identity for 2 2 matrices is the
matrix
1 0
a b
. Here,
0 1
c d
1 0
0 1
more complicated but is given by: A
a b
. Finding the inverse of a matrix is a lot
c d
a b
c d
A -1
d
1
det A c
b
a
2
4
Example 6 : Find the inverse of A
det
A -1
2
3
4
4
8 12
1 4 3
20 4 2
20
1
5
1
5
3
20
1
10
5 2
.
1 6
Example 7 : Find the inverse of A
det
A -1
5
2
1 6
1 6
32 1
30 2
2
5
32
3
16
1
32
1
16
5
32
3 2
Example 8 : Find the inverse of A
det
3 2
12 12
6 4
3
.
4
6 4
.
0
Since the det A = 0, then there is NO inverse for A. This is similar to the fact that the number
0 has no reciprocal.
Solving Systems of Equations with Matrices
In Algebra, to solve the equation ax
b, we multiply both sides of the equation by the reciprocal
b
of a.
a 1 ax a 1 b
x a1 b
a
Solving systems of equations using matrices is perhaps the best way of doing the problem.
ax by
e
cx dy
f
can be written as
In matrix terminology this is: A X
A -1 A X
A -1 B
X = A -1 B
a b
c d
x
y
e
f
B. To solve we multiply by A -1 which gives:
Example 9: Solve the system the following system of equations using matrices:
3x 2 y
x
y
3
1
6
2
1
x
y
6
2
2
A -1
det A = 5
1
5
1
5
1 1 2
5 1 3
2
5
3
5
1 2
6
2
5 5
y
1 3
2
0
5 5
Thus, x
2 and y 0.
x
Example 10: Solve the system the following system of equations using matrices:
3x 4 y
40
5 x 10 y
4
x
40
5
10
y
50
50
A -1
det A = -50
x
y
3
1
5
1
10
2
25
3
50
1
50
40
50
10
5
4
3
1
5
1
10
2
25
3
50
4
7
Example 11: Solve the system the following system of equations using matrices:
3x 2 y
6x 4 y
3 2
6 4
4
x
y
4
5
5
det A = 0
A -1 does not exist. This means that there is not a single solution to the
given problem. Either there are INFINITELY MANY solutions or NO solutions. If we
graph these two equations we can see that there are parallel. Thus, there is NO
solution to the above system of equations.