Download 4.5 Identify and Inverse Matrices

4.4 Identify and
Inverse Matrices
Algebra 2
Learning Target

I can find and use inverse matrix.
Introduction

There are certain properties of real
numbers that are related to special
matrices. Remember that 1 is the identity
for multiplication because 1 ∙ a = a ∙ 1 = a.
The identity matrix is a square matrix that,
when multiplied by another matrix, equals
that same matrix.
1 0
With a 2 x 2 matrices, 
is the identity matrix

0 1 
because
a b  1 0 a b 
 c d   0 1   c d 

 
 

and
1 0 a b  a b 
0 1   c d    c d 

 
 

The identity matrix is symbolized by I. In any identity
matrix, the principal diagonal extends from upper left to
lower right and consists only of 1’s
Identity Matrix for Multiplication(The
rule)

The identity matrix I for multiplication is a
square matrix with a 1 for every element of the
principal diagonal and a 0 in all other
positions.
3 2  1
3 2  1
Ex. 1: Find I so that  8 4 1 I   8 4 1




In order for you to multiply the matrices, remember that
the number of columns of the first matrix must equal the
number of rows in the second one.
The dimensions of the first matrix are 2 x 3. So I must
have 3 rows. Since all identity matrices are square, it
also has 3 columns. The principal diagonal contains 1’s.
Complete the matrix with 0’s
1
 1 0 0
 1   0 1 0

 


1 0 0 1
The 3 x 3 identity
matrix is
1 0 0
0 1 0


0 0 1
Next

Another property of real numbers is that any
real numbers is that any real number except 0
has a multiplicative inverse. That is 1/a is the
multiplicative inverse of a because a ∙ 1/a = 1/a
∙a = 1. Likewise, if matrix A has an inverse
named A-1, then A ∙ A-1= A-1∙ A = I. The
following example shows how a 2 x 2 matrix
can be found.
Inverse form for a 2 x 2
For any matrix M,
a b 
 c d  will have an inverse M-1


a b 
If an only if 

c d 
Then M-1 =
1  d  b


ad  bc  c a 
Ex. 3: If A =
 3 5 
 1  4


,find A-1 and check
your results
Compute the value of the
determinant.
Since the determinant
does not equal 0, A-1
exists.
 3 5 
 1  4  12  5  7


1  4  5
A  
7   1  3
1
Check:
1  4  5  3 5  1 12  5  20  20 1 0

 





7   1  3  1  4 7  3  3  5  12  0 1
Assignment
Pg. 227 #6-24
 Pg 233 12-30
