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. a23 a12 a31 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 Multiplication When multiplying matrices two matrices find the dimensions of each:

Is it possible to multiply these matrices 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?? x 2. x 1x3 and 3x3 1x3

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

Determinant of a Matrix 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 2x2

Find the determinant of the following 2x2 matrices:

Determinant of a 3x3 Matrix

Find the determinant of the following. -161 and 139

Matrix Equations

Matrix Equation Example

Solve each equation:

Inverse of Matrices

For matrices, there is no such thing as division 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

Using Inverses to Solve For X AX=C

Inverse Notation REMEMBER we denote inverse with a -1 power So the inverse of matrix A is A-1

Requirement to have an Inverse Matrix 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 Inverse When 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! 2nd  Matrix  Edit Put in your matrix 2nd  Matrix  NAME Get your matrix X-1

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

Solve each matrix equation. Solutions: