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Matrices
Addition & Subtraction
Scalar Multiplication & Multiplication
Determinants
Inverses
Solving Systems – 2x2 & 3x3
Cramer’s Rule
Matrices
 Matrix – A rectangle array of terms (elements)
arranged in columns and rows. A matrix with
m rows and n columns is called an m x n
matrix, (read m by n matrix).
 Matrices are also used to determine solutions
for multiple variable linear equations. This
technique can be used as an alternative to
elimination or substitution methods.
Matrices
a11 a12 a13
a21 a22 a23
a31 a32 a33
3 x 3 Matrix
The first number indicates the row
(horizontal) and the second number
indicates the column number (vertical).
Equal Matrices – Two matrices are equal if and only they
have the same dimensions and are equal element by
element.
This expression states that
Y
X
=
2x – 6
2y
Y = 2x – 6 and x = 2y. Using the
substitution method, we see that
Y = 2(2y) – 6 and so y = 2, x = 4.
Matrices
Addition of Matrices – The sum of two m x n matrices is a
m x n matrix in which the elements are the sum of the
corresponding elements of the given matrices.
A = -2 0 1
0 5 -8
A+B
=
B
=
-6 7 -1
Solve for A + B.
4 -3 10
-2 + (-6)
0+4
0 + 7 1 + (-1)
5 + (-3)
-8 + 10
A+B
=
-8 7 0
4 2 2
Matrices
Subtraction of Matrices – The difference of two m x n
matrices is equal to the sum A + (-B) where (-B) is the
additive inverse of B.
A = -2 0 1
0 5 -8
A-B
=
B
=
-6 7 -1
Solve for A - B.
4 -3 10
-2 - (-6)
0-4
0-7
5 - (-3)
1 - (-1)
-8 - 10
A-B
=
4 -7 2
-4 8 -18
Matrices
Scalar Product – The product of a scalar k and an m x n
matrix A is an m x n matrix denoted by kA. Each element of
kA equals k times the corresponding element of A.
A = -2 0 1
0 5 -8
kA
=
k
=
5(-2)
5(0)
Solve for kA.
5
5(0 ) 5(1)
5(5) 5(-8)
kA
=
-10 0
5
0 25 -40
Matrices

Multiplication – The column value of the first matrix must be the
same as the row number of the second matrix in order for
multiplication to occur.
 2 4
 3 1 
 2 4   3 1 
A
B




A * B= 


1 3
 4 2 
1
3
4

2

 

 (2* 3)  (4*4) (2*1)  (4* 2)  10 6 



(1*

3)

(3*4)
(1*1)

(3*

2)
9

5

 

A is a 2 x 2 matrix and B is also a 2 x 2 matrix.
Because the column number for A is a 2 and the
row number for B is a 2, multiplication is possible.
Matrices

Multiplication
 1 4 
 5

A
&B  

2

3

2




 (1*5)  (4* 2)
  13
A B  

 (2*5)  ( 3* 2)
  16
 5
  1 4 
B A 


 2   2 3 



A is a 2 x 2 matrix and B is a 2 x 1 matrix. Because the column
number for A is a 2 and the row number for B is a 2, multiplication is
possible which means that A x B is possible. However, B x A is not
possible because the column number of B is 1 and the row number of
A is 2.
Matrices

Multiplication of 3 x 3 with a 3 x 3.
 R1C1

 R2C1
RC
 3 1
R1C2
R2C2
R3C2
R1C3 

R2C3 
R3C3 
Rows of A times Columns of B with three
sums for each position
(2*1)  ( 1*2)  (3*3)
(2*4)  (1* 1)  (3*2)   11 9 15 
 2 1 3   3 1 4   (2*3)  (1*1)  (3*2)

 
 
 

 3 4 2    1 2 1   (3*3)  (4*1)  (2*2) (3*1)  (4*2)  (2*3) ( 3*4)  (4* 1)  (2*2)    9 1 20 
 4 5 1   2 3 2   (4*3)  (5*1)  (1* 2) (4*1)  (5*2)  (1*3)
(4* 4)  (5* 1)  (1*2)   9 9 19 

 
 
Determinants and Inverses

A determinant is a square array of numbers (written within
a pair of vertical lines) which represents a certain sum of
products.


Calculating a 2 × 2 Determinant
In general, we find the value of a 2 × 2 determinant with elements
a, b, c, d as follows:
We multiply the diagonals (top left × bottom right first), (bottom left x
top right) then subtract the first product minus the second.
det
a b
c d
=
a b
c d
=
ad - cb
Determinants & Inverses
The minors of the first row times the first row coefficients with the
alternating sign changes are used to find the determinant of the
matrix.
4 2 1
 3 1



3
3

1


4





2
0


 5 2 0 


4 2 1
 3 1


 3 3 1  2  5 0 


 5 2 0 


4 2 1
 3 3 



3
3

1


1




5

2


 5 2 0 


Determinants & Inverses

A 3 x 3 matrix determinant requires the use of the minors of the

top row of terms. It also includes the placement of alternating
(+) and (-) signs as operators with the minors.
The minors are the four elements that are not included in the
row or column of the element from the first row that is the
coefficient of the minor.
+
_
+
 2 3 1 
 1 4   2 4 
 2 1 


 2 1 4   2  3 6   3  5 6   (1)  5 3  

 



 5 3 6 


2(6  12)  3(12  20)  1(6  5)  12  24  1  37
Determinants & Inverses

The inverse of a matrix, A-1, is a matrix such that the product of
a matrix and its inverse will always result in the formation of the
identity matrix.
 2 1 
A

 5 6 
 6 1 
7 7 
1
A 

 5 2 


7 7 
1 0
Identity  

0 1
6
5
1
2   7
 6 1  
(2

)

(

1

)
(2

)

(

1

)
 2 1   7 7  
7
7
7
7  7




5

6
5

2
6
5

1

2
  (5  )  (6  ) (5  )  (6  )   0

 

 
 
7
7
7
7  7
7 7  
0
7  1 0


7  0 1

7
Determinants & Inverses
 The process to solve for the inverse of a 2 x 2 matrix is as
follows:
 Solve for the determinant of the matrix. This is done with
cross-multiplication and subtraction.
 Solve for the transpose of matrix A; AT : This is done by
reversing the order of the first and fourth term of the matrix
and multiplying the second and third term by (-1).
 The product of 1/det A and the AT matrix will create the A-1.
Ex.
1 2

 ;det A  (1* 4)  (2*3)  4  6  2
3 4
Determinants & Inverses
 4 2 
T

A


 3 1 
1 1   x
 7

*

 


 2 4   y
 8

det A  (1* 4)  (2*1)  4  2  6
4 1 
1  4 1  6 6 
  A1


6  2 1   2 1 


6 6 
1
4 1 
 4
*7

*8
 6 6  7   6
6



2

1
8

 
  2 *7   1 *8



6
6 6 
6
  36
  6

  6
 
  6

 6

 1





 2
4

2


1

A1 

 3
2  3 1  
 2
1 

1 

2 
Determinants & Inverses

Ex:
The product of the inverse of a matrix and the constant matrix of
a system will yield the values of the variables in the system.
1 1   x

 *
 2 4   y
 7

 8



det A  (1* 4)  (2*1)  4  2  6
4 1 
1  4 1  6 6 
  A1


6  2 1   2 1 


6 6 
1
4 1 
 4
*7

*8
 6 6  7   6
6



2

1
8

 
  2 *7   1 *8



6
6 6 
6
  36
  6

  6
 
  6

 6

 1





Cramer’s Rule
Cramer’s Rule begins with the solving
of the determinant for the system
followed by the determinants for each
of the variables within the system.
The determinant for each of the
variables is calculated by first
substituting the solution column
values for the variable column values
and then worked on as a 2 x 2
matrix. This method is used to solve
3x3 matrices instead of trying to
solve for the A-1 of the matrix.
Cramer’s Rule (continued)
Cramer’s Rule (conclusion)