MATH 166 Spring 2016
5.3
c
Wen
Liu
5.3 Inverse of a Square Matrix
Definition:
• A matrix is called a square matrix if the number of rows equals the number of columns.
• The n × n identity matrix, In , is the square matrix of order n × n with ones down the main
diagonal and zeros elsewhere.
• The n × n matrix B is said to be the inverse to the n × n matrix A if
AB = BA = In
If the inverse matrix B exists, we write B = A−1 .
Remark: Do not think that the inverse matrix A−1 can be written as 1/A. We never divide matrices.
Solution of AX = B: Let A be a square matrix of order n and B a column matrix of order n. If
A−1 exists, then the linear system of equations
AX = B
has the unique solution
X = A−1 B
Remark: If A−1 does not exist, then the system of equations does not have a single, unique solution.
The system must have a parametric solution or no solution in this case.
Technology Corner: To compute A−1 : enter matrix A. On the home screen, enter the matrix
name [A], then press the inverse button x−1 .
Examples:
1. Determine if the matrices are inverses to each other by showing if their product is the identity
matrix.
4 1
3 1
A=
,B =
11 3
−11 4
2. Find the inverse of the matrix, if it exists. (If an answer does not exist, enter DNE.)
8 −1
A=
7 5
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MATH 166 Spring 2016
c
Wen
Liu
5.3
3. Find the inverse of the matrix, if it exists. (If an answer does not exist, enter DNE.)
A=
4 2
6 3
4. Consider the system of equations given.
−2x − y + 3z = 3
x + y + 2z = 3
2x + 2y + 3z = −3
(a) Write a matrix equation (in the form AX = B) that is equivalent to the system of linear
equations. (Use the order of the equations to write the matrix.)
(b) Solve the system.
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