Matrices

Matrix Operations A matrix is a rectangular arrangement of numbers in rows and columns. A = 2 rows 3 columns In the matrix above, the dimensions are 2 x 3 & the numbers in the matrix are its entries. Two matrices are equal if their dimensions are the same and the entries in the corresponding positions are equal. 6 2 −2 0 −1 5

Special Matrix Name Description Example Row Matrix A matrix with only 1 row 3 −2 0 4 Column Matrix A matrix with only 1 column 1 3 Square Matrix A matrix with the same number of rows and columns 4 −1 5 2 0 1 1 −3 6 Zero Matrix A matrix whose entries are all zeros 0 0 0 0 0 0

Comparing Matrices Compare the following Matrix to determine if they are equal. A) 5 0 − 4 4 3 4 and 5 0 −1 0.75 B) −2 6 0 −3 and −2 6 3 0

Adding & Subtracting matrices Add/subtract the corresponding entries together. They must have the same dimensions in order to combine. Example: Perform the indicated operation, if possible. A) 3 −4 7 + 1 0 3 b) 8 3 4 0 − 2 −7 6 −1 c) 2 0 3 4 + 1 5

Scalar Multiplication In matrix algebra, a real number is often called a scalar. When you multiply a matrix by a scalar, you multiply each entry by the scalar.

Multiplying a matrix by a scalar Note: multiply a scalar through before using addition or subtraction. Perform the indicated operation(s), if possible. A) 3 −2 0 4 −7 b) −2 1 −2 0 3 −4 5 + −4 5 6 −8 −2 6

Solving a matrix Equation You will use what you know about matrix operations and matrix equality to solve a matrix equation. Solve the matrix equation for x and y: 3𝑥 −1 8 5 + 4 1 −2 −𝑦 = 26 0 12 8

Multiplying Matrices The product of two matrices A and B is defined provided the number of columns in A is equal to the number of rows in B. If A is an m x n matrix and B is an n x p matrix, then the product AB is an m x p matrix.

Describing Matrix Products State whether the product AB is defined. If so, give the dimensions of AB. 1) A: 2x3, B: 3x4 2) A: 3x2, B: 3x4

Finding the product of two matrices Find AB if A = −1 5 5 2 0 −4 and B = 4 −3 6 8

Finding the product of two matrices Find AB if A = −2 3 1 −4 6 0 and B = −1 3 −2 4

Finding the Product of Two matrices If A = 3 2 −1 0 and B = 1 −4 2 1 1) Find AB 2) Find ba

Using Matrix Operations If A = 2 1 −1 3 and B = −2 0 4 2 and C = 1 1 3 2 1) Simplify a(b + c) 2) Simplify AB + ac

Using matrix multiplication in real life Matrix multiplication is useful in business applications because an inventory matrix, when multiplied by a cost per item matrix, results in a total cost matrix. Example: Two softball teams submit equipment lists for the season. Each bat costs $21, each ball costs $4, and each uniform costs $30. use matrix multiplication to find the total cost of equipment for each item. Women’s Team Men’s Team 12 bats 15 bats 45 balls 38 balls 15 uniforms 17 uniforms

Each bat costs $21, each ball costs $4, and each uniform costs $30 Each bat costs $21, each ball costs $4, and each uniform costs $30. use matrix multiplication to find the total cost of equipment for each item. Women’s Team Men’s Team 12 bats 15 bats 45 balls 38 balls 15 uniforms 17 uniforms