Investigation: Inverse Matrices
Solving matrix equations that involve addition/subtraction and scalar is very
straightforward, since the operations themselves is simple.
However, solving equations that involve matrix multiplication is much more involved,
since matrix multiplication is complex. So, here’s how we are going to solve equations
such as
 2  5
  2
X  
1

3

2
Step 1 The symbol x-1 indicates the multiplicative inverse of a number or object.
1
1
1
Example: 3-1 = . This means that the multiplicative inverse of 3 is because  3  1 ,
3
3
3
so they cancel each other out.
 2  5
Since we are trying to undo the matrix, 
, we are not dividing, but instead, we
3 
1
are going to multiply by its inverse. On your calculator, find the inverse.
 2  5
1
3 

1
 __

 __
__ 
__ 
Step 2 Since matrix multiplication is no commutative, the order in which we multiply
the matrices is very important. The inverse will always go first, so the equation will look
like this:
AX= B
A-1AX = A-1B
We need to multiply by the inverse to both sides.
notice how the inverse is on the left of the everything on
each side.
X = A-1B
A-1A = I (identity), i.e. they cancel out.
So fill out the following for our matrix equation.
 2  5
  2
X  
1

3

2
 __
 __

__   2  5
 __
X

 __
__   1
3 

 __
X 
 __
Use matrix
multiplication to
find your
answer!
__   2
__   2 
__   2
__   2 
*Note: what would
happen if you put the
inverse on the other side
of the 2x1 matrix?
 __ 
X  
 __ 
Step 3 Your Turn!
5  3
5
Solve the matrix equation 
X  

 4  2
10
Step 4 Try it with a 3x3. What will the dimensions of your answer be?
2
0 0
 6 


Solve 1 3  2 X   11
1  2 1 
 8 
Step 5 Matrix equations allows us another way of solving systems of equations, such as
2 x  3 y  11

x  2 y  6
So we need to convert these equations into a matrix equation. Note: Equations must be
written in standard form first (x then y equals constant)
The matrix equation will be set up as following
[Coefficients][Variables]=[Constants]
[2x2]
[2x1]
[2x1]
Convert the above example into a matrix equation.
x’s
Coefficients for the first equation
Coefficients for the second.
 __
 __

y’s
x then y
__   __   __ 
 



__   __   __ 
Constant of first
Constant of second
Step 6 Now solve the equation just like before. Your answer will represent the x value
and y value. Check your answer and try solving it using elimination and substitution to
see if it works as well.
Step 7 Try 2 more! Solve:
5a  3b  7
a. 
3a  2b  5
x  y  z  2

b. 2 x  y  5
 x  3 y  3z  14
