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Warm up

Warm up. Solve this system of equations using an augmented matrix: 2x − 3y − z = −7 −x + 2y + z = 6 9x − 4y + 4z = 5. Warm-Up. Give the dimensions of each matrix. 2). 1). Identify the entry at each location of the matrix below. 3) b 12. 4) b 21. 5) b 32.

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Warm up

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  1. Warm up Solve this system of equations using an augmented matrix: 2x − 3y − z = −7 −x + 2y + z = 6 9x − 4y + 4z = 5

  2. Warm-Up Give the dimensions of each matrix. 2) 1) Identify the entry at each location of the matrix below. 3) b12 4) b21 5) b32

  3. Lesson 11-2 Matrix Operations Objective: To learn to add matrices and perform scalar multiplication.

  4. Adding Matrices • In order to add matrices each one must have the same number of rows and also the same number of columns. (You can add a 3 x 2 matrix to another 3 x 2 matrix, but not to a 1 x 5 matrix). • Matrix equality occurs when 2 matrices have the same dimensions and the same entries.

  5. Adding Matrices • To add matrices that are the same size, add the elements in each position.

  6. Adding Matrices • Example: 2 1 1 5 2+1 1+5 3 6 0 -1 + 2 0 = 0+2 -1+0 = 2 -1 3 4 -1 1 3-1 4+1 2 5

  7. Adding Matrices • Try: -5 2 10 3 7 -1 + -2 4 = 8 9 -3 0

  8. Adding Matrices • Answer: -5 2 10 3 -5+10 2+3 5 5 7 -1 + -2 4 = 7-2 -1+4 = 5 3 8 9 -3 0 8-3 9+0 5 9

  9. Scalar Multiplication of Matrices • The first type of multiplication we will investigate is called scalar multiplication. • In scalar multiplication each element in a matrix is multiplied by a number, called a scalar.

  10. Scalar Multiplication of Matrices Example: 11 -5 2 x 11 2 x -5 22 -10 2 -9 6 = 2 x -9 2 x 6 = -18 12 -4 3 2 x -4 2 x 3 -8 6 scalar

  11. Scalar Multiplication of Matrices • Try: -4 -5.3 2 3 3.1 0 6 = 1/3 -9 1

  12. Scalar Multiplication of Matrices • Answer: -4 -5.3 2 -12 -15.9 6 3 3.1 0 6 = 9.3 0 18 1/3 -9 1 1 -27 3

  13. Warm up 1. 2. 3.

  14. Multiplication of Matrices • Multiplying a matrix by another matrix is a little more complicated. When multiplying 2 matrices (matrix A and matrix B.) The number of columns in matrix A must be equal to the number of rows in matrix B. • If this is not true, then the matrices can not be multiplied.

  15. Multiplication of Matrices • Example: Matrix A Matrix B Three columns three rows 7 2 5 -3 2 3 1 6 1 These two matrices can be multiplied together. In this case we have a 2x3 matrix being multiplied by a 3x1 matrix. The result will be a 2x1 matrix.

  16. Multiplication of Matrices In order to find the product, multiply each element in the first row of A with the corresponding element in the first column of B. Then you add each of these answers together and this gives you just one element of the answer.

  17. Multiplication of Matrices Example: 7 2x7 + 5x2 + (-3x1) 21 2 5 -3 2 = 3x7 + 1x2 + 6x1 = 29 3 1 6 1

  18. Multiplication of Matrices This can also be done when the second matrix has more than one column. Multiplying a 2x3 matrix with a 3x2 matrix will give you a 2x2 matrix.

  19. Multiplication of Matrices Example: 7 3 2 5 -3 2 9 = 3 1 6 1 2 1st row x 1st column 1st row x 2nd column 2x7 + 5x2 + (-3x1) 2x3 + 5x9 + (-3x2) 21 45 3x7 + 1x2 + 6x1 3x3 + 1x9 + 6x2 = 29 30 2nd row x 1st column 2nd row x 2nd column

  20. Multiplication of Matrices • Now try one: 1 0 2 1 0 3 1 0 = 2 0 5 4

  21. Multiplication of Matrices • Now try one: 1 0 2 1 9 0 3 1 0 = 4 2 0 5 4 22

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