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Name: ______________________________________ Period ______ Sec6-3 Matrix Equations
In this lesson you will: (1)
(2)
(3)
(4)
find an identity matrix
find the inverse of a matrix
write matrix equations
use inverse matrices to solve matrix equations
Start by reviewing what you’ve learned about matrices:
Scalar Multiplication and Matrix Multiplication
(a) 8 10   5 5 
(b) 8
9 3    2 3
9

 


0 6   1 2 
 0
(c)
5 2 
6  1 1
 4 7 
(d)
Adding, Subtracting,
10   5 5 
3    2 3
6   1 2 
1 4 
1 2 3  

4 5 6 2 5

 3 6


Find the missing values. SHOW WORK
 4 a  1  10 
b 5   2   20

   
Identity Matrix:
In earlier math courses, you learned that the number 1 is the multiplicative identity.
This means that when you multiply any real number by 1, the number does not change.
Similarly, when you multiply a matrix by an identity matrix, the matrix does not change.
 2 1
a b 
Find the identity matrix for matrix 
means
you
need
to
find
the
matrix

 c d  so that
 4 3


2
1
a
b
2
1





The product 
equals 




 4 3  c d 
 4 3
Find the values for a, b, c, d by using matrix multiplication.
In general, an identity matrix is a square matrix with 1’s along the main diagonal from top
left to bottom right, and 0’s for all the other entries. We use and “I” to name the identity
matrix.
1 0 0 0 
1 0 0 
0 1 0 0 
1 0 
0 1 0 


I= 
or
I=
or
I=
and so on …



0 0 1 0 
0 1 
0 0 1 


 0 0 0 1
The Inverse Matrix:
In earlier math course you also learned that every nonzero real number has a
multiplicative inverse, the number you multiply it by to get the multiplicative identity, 1.
For example: The multiplicative inverse of 4 is ¼ because (4)(1/4) = 1
Similarly, SOME (but not all) SQUARE matrices have an inverse matrix. We write the inverse
of a matrix A as A-1. when you multiply a matrix on either side by its inverse matrix you get
the identity matrix.
This means that: A A-1=I and also A-1A=I
 2 3   4 3 
(a) Are these two matrices inverses of each other? 


1 4   1 2 
7 5   2 5 
(b) Are these two matrices inverses of each other? 


 3 2  3 7 
 2 1
(c) Find the inverse of matrix A= 

 4 3
 2 1  a b  1 0 
You will need to solve for a, b, c, d by using matrix multiplication: 
  c d  = 0 1 
4
3



 
Confirm that you have the correct inverse matrix by using your calculator to multiply
matrix A with it’s inverse.
Another way to confirm that you found the correct inverse is to let the calculator find the
1
 2 1
-1
inverse matrix: A = 

 4 3
Write a matrix equation to represent a system of equation:
x  2y 1
3x  4 y  2
Coefficient






Variable






 
 

 
 
 
Constant






Solve the matrix equation by using the Inverse Matrix:
Steps:
(1) Find the inverse of the coefficient matrix.
(2) Multiply both sides of the equation by the Inverse Matrix
(3) Simplify to solve for the variable matrix.
Practice Problems
2 x  5 y  8
(1) Write a matrix equation and then solve by using the Inverse Matrix. 
 4x  y  6
 2x  y  2z  1

(2) Write a matrix equation and then solve by using the Inverse Matrix. 6 x  2 y  4 z  3
 4 x  y  3z  5

(3) Find a, b, c such that the graph of
(2, -5), (4, 21) and (-2, 15)
y  ax 2  bx  c
passes through the points: