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Understanding Matrix Multiplication: Concept, Properties, and Applications

Learn about matrix multiplication, its properties, and real-world applications. Explore how to multiply matrices, associative property, scalar multiplication, and more.

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Understanding Matrix Multiplication: Concept, Properties, and Applications

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  1. MULTIPLYING TWO 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.

  2. MULTIPLYING TWO MATRICES ABAB 4X 33 X54X5 4 rows 3 rows 3 columns 5 columns

  3. 4 rows 5 columns MULTIPLYING TWO MATRICES ABAB 4X 33 X54X5 4 rows 5 columns

  4. MULTIPLYING TWO MATRICES If A is a 4X3matrix and B is a 3X5matrix, then the product ABisa 4X5 matrix.

  5. MULTIPLYING TWO MATRICES A  B  AB mXnn XpmXp m rows n rows n columns p columns

  6. m rows p columns MULTIPLYING TWO MATRICES ABAB mX nn XpmXp m rows p columns

  7. MULTIPLYING TWO MATRICES If A is an mXnmatrix and B is an nXpmatrix, then the product AB isan mXp matrix.

  8. Finding the Product of Two Matrices –2 3 1 –4 6 0 –1 3 –2 4 Find AB if A = and B = SOLUTION Because A is a 3X 2matrix and B is a 2 X2matrix, the product AB is defined and is a 3X2matrix. To write the entry in the first row and first column of AB, multiply corresponding entries in the first row of A and the first column of B. Then add. Use a similar procedure to write the other entries of the product.

  9. Finding the Product of Two Matrices –2 3 1 –4 6 0 –13 –24 (–2)(–1) + (3)(–2) (–2)(3) + (3)(4) (1)(–1) + (–4)(–2)(1)(3) + (–4)(4) (6)(–1) + (0)(–2) (6)(3) + (0)(4) ABAB 3X22X23X2

  10. Finding the Product of Two Matrices –2 3 1 –4 6 0 –13 –24 (–2)(–1) + (3)(–2) (–2)(3) + (3)(4) (1)(–1) + (–4)(–2)(1)(3) + (–4)(4) (6)(–1) + (0)(–2) (6)(3) + (0)(4) ABAB 3X22X23X2

  11. Finding the Product of Two Matrices –2 3 1 –4 6 0 –13 –24 (–2)(–1) + (3)(–2) (–2)(3) + (3)(4) (1)(–1) + (–4)(–2) (1)(3) + (–4)(4) (6)(–1) + (0)(–2) (6)(3) + (0)(4) ABAB 3X22X23X2

  12. Finding the Product of Two Matrices –4 6 7 –13 –6 18 (–2)(–1) + (3)(–2) (–2)(3) + (3)(4) (1)(–1) + (–4)(–2) (1)(3) + (–4)(4) (6)(–1) + (0)(–2) (6)(3) + (0)(4) ABAB 3X22X23X2 

  13. CONCEPT SUMMARY PROPERTIES OF MATRIX MULTIPLICATION MULTIPLYING TWO MATRICES Matrix multiplication is not, in general, commutative. Let A, B, and C be matrices and let g be a scalar. ASSOCIATIVE PROPERTY OF MATRIX MULTIPLICATION A(BC) = (AB)C A(B + C) = AB + AC LEFT DISTRIBUTIVE PROPERTY (A + B)C = AC + BC RIGHT DISTRIBUTIVE PROPERTY ASSOCIATIVE PROPERTY OF SCALAR MULTIPLICATION g(AB) = (gA)B = A(gB)

  14. Inventory matrix Cost per item matrix Total cost matrix 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 total cost matrix. = • mXn nXp mXp For the total cost matrix to be meaningful, the column labels for the inventory matrix must match the row labels for the cost per item matrix.

  15. Using Matrices to Calculate the Total Cost SPORTSTwo 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 team.

  16. Using Matrices to Calculate the Total Cost SOLUTION Write the equipment lists and costs per item in matrix form.Use matrix multiplication to find the total cost. Set up matrices so that columns of the equipment matrix match rows of the cost matrix. COST EQUIPMENT Dollars Bats Balls Uniforms 21 Women’s team 12 45 15 Bats 4 15 38 17 Men’s team Balls 30 Uniforms

  17. Using Matrices to Calculate the Total Cost 21 12 45 15 4 15 38 17 30 Total equipment cost for each team can be obtained by multiplying the equipment matrix by the cost per item matrix. The equipment matrix is 2 X3 and the cost per item matrix is 3 X 1. Their product is a 2 X1 matrix. 12(21) + 45(4) + 15(30) 882 = = 15(21) + 38(4) + 17(30) 977

  18. Using Matrices to Calculate the Total Cost TOTAL COST Dollars The labels of the product are: 882 Women’s team 977 Men’s team The total cost of equipment for the women’s team is $882,and the total cost of equipment for the men’s team is $977.

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