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MATRICES

- MATRIX OPERATIONS

About Matrices

- A matrix is a rectangular arrangement of numbers

in rows and columns. Rows run horizontally and

columns run vertically. - The dimensions of a matrix are stated m x n

where m is the number of rows and n is the

number of columns.

Equal Matrices

- Two matrices are considered equal if they have

the same number of rows and columns (the same

dimensions) AND all their corresponding elements

are exactly the same.

Special Matrices

- Some matrices have special names because of what

they look like. - Row matrix only has 1 row.
- Column matrix only has 1 column.
- Square matrix has the same number of rows and

columns. - Zero matrix contains all zeros.

Matrix Addition

- You can add or subtract matrices if they have the

same dimensions (same number of rows and

columns). - To do this, you add (or subtract) the

corresponding numbers (numbers in the same

positions).

Matrix Addition

Example

Scalar Multiplication

- To do this, multiply each entry in the matrix by

the number outside (called the scalar). This is

like distributing a number to a polynomial.

Scalar Multiplication

Example

Matrix Multiplication

- Matrix Multiplication is NOT Commutative! Order

matters! - You can multiply matrices only if the number of

columns in the first matrix equals the number of

rows in the second matrix.

Matrix Multiplication

- Take the numbers in the first row of matrix 1.

Multiply each number by its corresponding number

in the first column of matrix 2. Total these

products.

The result, 11, goes in row 1, column 1 of the

answer. Repeat with row 1, column 2 row 1

column 3 row 2, column 1 ...

Matrix Multiplication

- Notice the dimensions of the matrices and their

product.

3 x 2

2 x 3

3 x 3

__

__

__

__

Matrix Multiplication

- Another example

3 x 2

2 x 1

3 x 1

Matrix Determinants

- A Determinant is a real number associated with a

matrix. Only SQUARE matrices have a determinant. - The symbol for a determinant can be the phrase

det in front of a matrix variable, det(A) or

vertical bars arounda matrix, A or .

Matrix Determinants

To find the determinant of a 2 x 2 matrix,

multiply diagonal 1 and subtract the product of

diagonal 2.

Matrix Determinants

To find the determinant of a 3 x 3 matrix, first

recopy the first two columns. Then do 6 diagonal

products.

Matrix Determinants

The determinant of the matrix is the sum of the

downwards products minus the sum of the upwards

products.

(-8) - (94) -102

Identity Matrices

- An identity matrix is a square matrix that has

1s along the main diagonal and 0s everywhere

else.

- When you multiply a matrix by the identity

matrix, you get the original matrix.

Inverse Matrices

- When you multiply a matrix and its inverse, you

get the identity matrix.

Inverse Matrices

- Not all matrices have an inverse!
- To find the inverse of a 2 x 2 matrix, first find

the determinant. - If the determinant 0, the inverse does not

exist! - The inverse of a 2 x 2 matrix is the reciprocal

of the determinant times the matrix with the main

diagonal swapped and the other terms multiplied

by -1.

Inverse Matrices

- Example 1

Inverse Matrices

- Example 2

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