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PreAP Algebra II
Topic 4-5 Solving Systems Using Matrix Inverses
What is the definition of a multiplicative inverse?
According to this, there should be inverses for matrices as well. However, a
matrix can have an inverse only if it is a square matrix. But not all square
matrices have inverses. If the product of the square matrix A and the square
matrix A-1 is the identity matrix I, then AA-1 = A-1A = I, and A-1 is the
multiplicative inverse matrix.
Determine if the following matrices are inverses.
 1 0 2 
 0.2 0 0.4 
2 3 
 10 6 
1) 
and 
2)  4 1  1 and  1.2 1  1.4 






 7 10 
 7  4
 2 0 1 
 0.4 0 0.2 
Find the inverse of the matrix using Augmented Matrices.
 4 3
3) 

2 1 
4 2 
4) 

 5 2 
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PreAP Algebra II
Topic 4-5 Solving Systems Using Matrix Inverses (cont)
The inverse of a matrix can be used to solve a system of equations. This process
is similar to solving an algebraic equation such as 5x = 20 by multiplying each
side by 1/5, the multiplicative inverse of 5.
To solve systems of equations with the inverse, you first write the matrix
equation AX = B, where A is the coefficient matrix, X is the variable matrix,
and B is the constant matrix.
 x y 8
The matrix equation representing 
is shown.
2
x

y

1

A 
1 1
 2 1
 
Coefficient matrix A
X = B
 x  8
 y  = 1 
   
Variable matrix X Constant matrix B
To solve AX = B, multiply both sides by the inverse A-1 so:
AX = B
-1
A AX = A-1B
1X = A-1B
because the product of A-1 and A is 1
X = A-1B
5) Let’s solve the above matrix equation
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PreAP Algebra II
Topic 4-5 Solving Systems Using Matrix Inverses (cont)
Write the matrix equation for the system and solve.
 3x  4 y  z  1

6)  x  3 y  z  2

2 x  y  2 z  1
Assign: Pg. 282-283: #14 – 24, 27
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