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Math 166-copyright Joe Kahlig, 09C
Section 5.3: Inverse of a Square Matrix
From section 5.2 we saw that we could express this system of equations as a matrix equation.
3x + 2y = 7
x + 4y = 10
"
3
1
2
4
# "
x
y
#
=
"
7
10
#
Any system of equations can be expressed as a matrix equation of the form AX=B, where A is the
coefficient matrix, X is the variable matrix, and B is the answer matrix. We would like to be able to
solve the equation for X.
Definition: The inverse of matrix A, denoted A−1 , is a matrix that has the following property:
AA−1 = I and A−1 A = I. From this property we see that A must be a square matrix in order to have
an inverse.
Example: Solve the matrix equation for J. Assume all matrices are square and all inverses are possible.
A) 3B + JA = D
B) J + AJ = B
C) J + JA = B
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Math 166-copyright Joe Kahlig, 09C
Exampe: If possible, compute the inverse to these matrices.

2

A) A =  -4
3
B) B =
"
1
2

4
-6
5
0
3

2

C) C =  -1
3
-4
3
-7
-2

1 
-1
2
6
#

12

5 
7
Example: Are these matrices inverses?
A=
"
4
2
5
3
#
B=
"
1.5
-1
-2.5
2
#
Example: Solve for x and y.
4x + 5y = 20
2x + 3y = 40
Example: Solve for x, y, and z.
x − y + 3z = 2
2x + y + 2z = 2
2x + 2y − z = −3
Example: Solve for x, y, and z.
x + 2y + 3z = 2
2x + 5y + z = 1
3x + 7y + 4z = 3
Other uses of Inverses/Matrices: Cryptography, Leontief Input-Output Models(Economics),
Markov Processes, Game Theory.