Example 1 Compare Matrices The table shows the

Example 1 Compare Matrices The table shows the number of items sold during the first day of a class fundraiser. Tell whether the matrix is equal to the fundraiser matrix. Explain. b. B = = a. A 10 15 8 6( 2 ) 14 5 10 15 16 6 14 5

Example 1 Compare Matrices ANSWER = The element 6 ( 2 ) is equal to 12. So, matrix A is equal to the fundraiser matrix because all corresponding elements are equal. b. B = a. A 10 15 8 6( 2 ) 14 5 10 15 16 6 14 5 Matrix B is not equal to the fundraiser matrix because the corresponding elements in the second row are not equal.

Checkpoint Compare Matrices Tell whether the matrices are equal. Explain. 1. 2 0 , 0 – 2 ANSWER Equal; their dimensions are the same and the elements in corresponding positions are equal.

Checkpoint Compare Matrices Tell whether the matrices are equal. Explain. 2. 1 – 1 1 (4 – 5) , 4 0 (3 – 1) 0 2 ANSWER Equal; their dimensions are the same and the elements in corresponding positions are equal.

Checkpoint Compare Matrices Tell whether the matrices are equal. Explain. 1 3. [ 1 3 ], 3 ANSWER Not equal; their dimensions are not the same.

Example 2 Add and Subtract Matrices Add or subtract, if possible. If not possible, state the reason. = = c. 3 – 11 1 – 12 – 2 15 2 +4 1 + ( – 1) 3+5 6 0 8 4 – 1 – 2 0 – ( – 2) 5 – 3 = b. 4 – 1 1 2 – – 2 3 0 5 = + 4 – 1 5 = a. 2 1 3 3 – 3 2 2 It is not possible to subtract the matrices. They have different dimensions.

Example 3 Use a Matrix Operation The number of items sold during the second day of the class fundraiser is shown. Second Day of Fundraiser Large T-shirts Sweatshirts Long sleeves 12 4 5 Extra Large 5 3 6

Example 3 Use a Matrix Operation a. Add the matrix for the second day of the fundraiser to the matrix for the first day of the fundraiser. First Day of Fundraiser Large T-shirts Sweatshirts Long sleeves 10 8 14 Extra Large 15 12 5 b. What is the total number of large, long sleeve shirts sold during the first two days of the fundraiser?

Example 3 Use a Matrix Operation SOLUTION a. Add corresponding elements in the matrices. 8 14 12 15 4 12 5 5 5 3 6 = 10 + + + = 12 5 4 3 5 6 + + 10 15 8 12 14 5 22 20 12 15 19 11 b. You need to find the element in the matrix from part (a) that corresponds to large, long sleeve shirts. The element in the third row and first column corresponds to large, long sleeve shirts. A total of 19 large, long sleeve shirts were sold.

Example 3 Use a Matrix Operation Large T-shirts 22 Sweatshirts 12 Long sleeves 19 Extra Large 20 15 11

Checkpoint Add and Subtract Matrices Add or subtract, if possible. If not possible, state the reason. 4. 3 0 1 3 + – 2 2 2 4 5. 3 4 7 + ANSWER 4 3 0 6 5 2 6 4 – 3 6 ANSWER Not possible; their dimensions are not the same.

Checkpoint Add and Subtract Matrices Add or subtract, if possible. If not possible, state the reason. 6. 5 2 4 3 – 3 0 1 2 ANSWER 1 – 1 2 – 2

Example 4 Multiply a Matrix by a Scalar – 1( 3 ) – 1( 0 ) – 3 0 a. – 1 = = ( ) – 1 – 2 – 1 1 – 2 1 2 – 1 4 2( – 1 ) b. 2 = 0 3 2( 0 ) – 2 8 2( 4 ) = 2( 3) 0 6

Example 5 Perform Multiple Operations A store sells small and large steel DVD racks with wooden bases. Each size is available in three types of wood: walnut, pine, and cherry. Sales of the racks for last month and this month are shown.

Example 5 Perform Multiple Operations Use the matrices to find the average monthly sales for the two month period. Last month ( A ) Small Large Walnut 12 11 Pine 28 20 Cherry 22 21 This month ( B ) Small Large Walnut 14 15 Pine 36 28 Cherry 20 17

Example 5 Perform Multiple Operations SOLUTION You can find the average monthly sales for the two month period by adding matrices A and B together and 1 multiplying the sum by. 2 1 1( A + B) = 2 2 12 11 28 20 22 21 1 = 2 26 26 64 48 42 38 + 14 15 36 28 20 17 Write the average. Add matrices A and B.

Example 5 Perform Multiple Operations = 13 13 32 24 21 19 Multiply each element by 1. 2

Example 6 Solve a Matrix Equation Solve the matrix equation for x. 2 x – 1 4 – 1 12 – 2 + = 3 0 3 15 6 15 SOLUTION To solve the matrix equation, add the matrices on the left side of the equation, equate the element involving x with its corresponding element, and solve for x. 2 x – 1 4 – 1 12 – 2 + = 3 0 3 15 6 15 Write original equation.

Example 6 Solve a Matrix Equation 2 x + 4 – 2 12 – 2 = 6 15 2 x + 4 = 12 2 x = 8 x =4 Add corresponding matrix elements. Equate corresponding elements 2 x + 4 and 12. Subtract 4 from each side. Divide each side by 2.

Checkpoint Perform Matrix Operations and Solve Matrix Equations Perform the indicated operation(s). 1 – 2 7. 2 3 5 8. 5 – 2 3 – 1 – 2 + 1 0 4 7 ANSWER 2 – 4 6 10 ANSWER – 15 5 25 35 Solve the matrix equation for x. 9. x 25 – 2 + 24 5 – 10 = 27 30 – 12 ANSWER 3

Checkpoint Perform Matrix Operations and Solve Matrix Equations Solve the matrix equation for x. 10. – 1 3 – 4 3 3 0 – = 3 x 7 6 5 18 2 ANSWER 8