Section 5.3
MATH 166:503
April 14, 2015
Topics from last notes: matrix, order, row and column matrices, square matrix, multiplication
by a scalar, addition of matrices, transpose of a matrix, zero matrix, properties of matrix addition, using matrix addition to solve word problems, matrix multiplication, properties of matrix
multiplication, identity matrix, using matrix multiplication to solve word problems
4
SYSTEMS OF LINEAR EQUATIONS AND MODELS
ex. (from WebAssign) The matrix below is in row-reduced form
x
M= 1
0
y
0
1
Solve the system.
(x, y, z, u) =
1
z
0
6
u
0
4
7
7
5
MATRICES
5.3
Inverse of a Square Matrix
A n x n matrix B is the inverse of an n x n matrix A if AB = BA = In . If the inverse exists, we
denote B = A−1 .
To find A−1 :
1. Form the augmented matrix [A|I]
2. Row reduce to get [I|B] if possible
3. Then B = A−1
ex. Find the inverse of A =
1 2
3 4
.
Check AA−1 = I.
2


−1 0 8
ex. A =  0 1 1 . Compute A−1 .
3 9 −6


−1 0
8
ex. A =  2 10 1 . Compute A−1 .
3 10 −7
If we write a system of equations as AX = B and A−1 exists,
If A−1 does not exist, the system has either no solutions or infinitely many solutions.
3
ex. Solve the following system using matrices
9a + 3b − c =10
1
−a + c =6
2
4a + b =5
ex. Solve the following system
3
2p + q =12
2
4
p + q =14
3
4
ex. Solve the following system
6x1 + 3x2 − x4 =18
x2 + x4 =0
3
x1 − x2 + 4x3 = − 4
2
5