Matrices

Addition & SubtractionScalar Multiplication & MultiplicationDeterminantsInversesSolving Systems – 2x2 & 3x3Cramer’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.

=Y

X

2x – 6

2y

This expression states that

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 + B-2 + (-6) 0 + 7 1 + (-1)

0 + 4 5 + (-3) -8 + 10

A= -2 0 1

0 5 -8

= -6 7 -1

4 -3 10

B Solve for A + B.

=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 - B-2 - (-6) 0 - 7 1 - (-1)

0 - 4 5 - (-3) -8 - 10

A= -2 0 1

0 5 -8

= -6 7 -1

4 -3 10

B Solve for A - B.

=A - B4 -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.

=kA5(-2) 5(0 ) 5(1)

5(0) 5(5) 5(-8)

A= -2 0 1

0 5 -8

= 5k Solve for kA.

=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

1 3 4 2

A * B= 2 4

1 3A

3 1

4 2B

(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&

2 3 2

( 1*5) (4* 2) 13

(2*5) ( 3* 2) 16

5 1 4

2 2 3

A B

A B

B A

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.

2 1 3 3 1 4 (2*3) ( 1*1) (3*2) (2*1) ( 1*2) (3*3) (2*4) ( 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)

4 5 1 2 3 2 ( 4*3) (5*1) (1* 2) ( 4*1) (5*2) (1*3) ( 4*

11 9 15

9 1 20

4) (5* 1) (1*2) 9 9 19

1 1 1 2 1 3

2 1 2 2 2 3

3 1 3 2 3 3

RC RC RC

R C R C R C

R C R C R C

Rows of A times Columns of B with three sums for each position

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.

a b

c ddet =

a b

c d= ad - cb

Determinants & Inverses

4 2 13 1

3 3 1 42 0

5 2 0

4 2 13 1

3 3 1 25 0

5 2 0

4 2 13 3

3 3 1 15 2

5 2 0

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.

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 11 4 2 4 2 1

2 1 4 2 3 ( 1)3 6 5 6 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.

1

2 1

5 6

6 1

7 75 2

7 7

1 0

0 1

6 1 6 5 1 2 7 0(2 ) ( 1 ) (2 ) ( 1 )

2 1 1 07 7 7 7 7 7 7 75 6 5 2 6 5 1 2 0 7 0 1

(5 ) ( 6 ) (5 ) ( 6 )7 7 7 7 7 7 7 7

A

A

Identity

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.

1 2;det (1*4) (2*3) 4 6 2

3 4A

Ex.

Determinants & Inverses

1

1 1 7*

2 4 8

det (1* 4) (2*1) 4 2 6

4 14 11 6 62 1 2 16

6 6

4 1 4 1 36*7 *8

7 66 6 6 6 62 1 8 2 1 6 1

*7 *86 6 6 6 6

x

y

A

A

4 2

3 1TA

1

2 14 21

3 13 12

2 2

A

Determinants & Inverses 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 7*

2 4 8

x

y

Ex:

1

det (1* 4) (2*1) 4 2 6

4 14 11 6 62 1 2 16

6 6

A

A

4 1 4 1 36*7 *8

7 66 6 6 6 62 1 8 2 1 6 1

*7 *86 6 6 6 6

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)