Unit 3: Matrices

Matrix: A rectangular arrangement of data into rows and columns, identified by capital letters.

Matrix Dimensions: Number of rows, m, by the number of columns, n. Read as “m by n” matrix. Also known as the order of a matrix.

◦RBC (ROWS BY COLUMNS)

Determine the dimensions of each matrix.

Elements

Matrix Element: Each number in a matrix, identified by its row and column.

Example: amn

Refers to the m-th row and n-th column

Example

Identify each element.

1. a23

2. a12

3. a31

4. a21

Adding and Subtracting Matrices

When matrices have the same dimension you add and subtract them by adding or subtracting each corresponding element.

Add or Subtract the following matrices:

Scalar Multiplication

Matrix MultiplicationWhen multiplying matrices two

matrices find the dimensions of each:

Is it possible to multiply these matrices? If so, what would the dimension of your answer matrix be?

1)

2)

Multiplying Matrices

Can the following Matrices be multiplied? If so, what dimensions will the product be??

1. x

2. x

How to multiply matrices

Multiply the elements of each row in the first matrix by the elements in each column of the second matrix

Add the products to get the new element.

Matrix Multiplication

Equivalent Matrices

DETERMINANT OF MATRICES

A special number that can be calculated from the matrix.

It tells us things about the matrix that are useful in systems of linear equations, in calculus, and more

The symbol for determinant is two vertical lines either side

Determinant of a Matrix

Determinant of a 2x2

Find the determinant of the following 2x2 matrices:

Determinant of a 3x3 Matrix

Find the determinant of the following.

MATRIX EQUATIONS

Matrix Equation Example

Solve each equation:

INVERSE OF MATRICES

For matrices, there is no such thing as division. You can add, subtract, and multiple matrices, but you cannot divide them.

There is a related concept called inversion

AX=C

Using Inverses to Solve For X

Inverse Notation

REMEMBER we denote inverse with a -1 power

So the inverse of matrix A is A-1

Requirement to have an InverseMatrix MUST be square, meaning it has the same number of rows and columns

Matrix MUST NOT have a determinant of zero.

Inverse exist?!

Does the inverse exist?!?!

Multiplying InverseWhen you Multiply a matrix A times it’s inverse, the Product is the Identity Matrix.

Identity Matrix is a square matrix where the top left to Bottom right diagonal are all ones, and everything else is a zero

Determine if the following matrices are inverses. 1.

2.

Finding the Inverse of a 2x2

IF

THEN

Find the inverse of the following matrix.

Use your calculator!

1. 2nd Matrix Edit2. Put in your matrix3. 2nd Matrix NAME4. Get your matrix5. X-1

The inverse of a matrix can be used when solving matrix equations.

For Matrices A and B, we can find Matrix X:

IF AX = B

THEN X = A-1B

*Solve for X:

X = A-

1B

You Try! Solve Each Matrix Equation:

Solutions: