MATRICES
INVERSE MATRICES TO SOLVE
LINEAR SYSTEMS
Identity Matrices

An identity matrix is a square matrix that
has 1’s along the main diagonal and 0’s
everywhere else.
1 0 0 
0 1 0 


 0 0 1 

1 0 
0 1 


When you multiply a matrix by the
identity matrix, you get the original
matrix.
Inverse Matrices
 When you multiply a matrix and its
inverse, you get the identity matrix.
 3 1  2 1   1 0 
 5 2   5 3   0 1 


 

Inverse Matrices
 Not all matrices have an inverse!
 To find the inverse of a 2 x 2 matrix,
first find the determinant.
a) If the determinant = 0, the inverse does
not exist!
 The inverse of a 2 x 2 matrix is the
reciprocal of the determinant times the
matrix with the main diagonal swapped
and the other terms multiplied by -1.
Inverse of a 2X2 Matrix

Inverse Matrices
 3 1
Example 1: A  

5
2


det(A) = 3(2) – (-5)(-1)
det(A)  6  (5)  1
1 2 1 2 1
A  


1  5 3  5 3
1
Inverse Matrices
Example 2:
 2 2 
B

5
4


det(B)  (8)  (10)  2
2 2
1 4
B  
 5

2  5 2    2
1
1
1
Solve a Matrix Equation

Solve a Matrix Equation

Solve a Matrix Equation

Example of Inverse Matrices

Example of Inverse Matrices

Basketball Problem

During the 2003-2004 NBA season, Dirk
Nowitzki of the Dallas Mavericks made a total
of 976 shots and scored 1680 points. His
shots consisted of 3-point field goals, 2-point
field goals, and 1-point free throws. He
made 135 more 2-point field goals than free
throws. Use an inverse matrix to find how
many of each type of shot he made.
Basketball Problem






x = 3-point field goals
y = 2-point field goals
z = 1-point free throws
x + y + z = 976 shots
3x + 2y + z = 1680 points
y – z = 135
Basketball Problem

Basketball Problem
