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14.1 Matrix Addition and Scalar Multiplication OBJ: To find the sum, difference, or scalar multiples of matrices
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EX: An automobile dealer sells four different models whose fuel economy is shown in the table below: This information can be displayed as a rectangular array of numbers enclosed by brackets, called a matrix (plural, matrices), usually labeled with a capital letter. Spts Car Se- dan Sta- tion Wag Van City Mpg 17 221716 High- way Mpg 233024 19
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sp se sw v M = 17 22 17 16 c 23 30 24 19 h Each number is an element (or entry) of the matrix. The dimensions are the number of rows and columns. Since M has two rows and four columns, M is a 2 x 4 matrix, denoted by M 2x4. It is a “driving-condition by model” matrix.
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If the rows and columns are interchanged, you get the transpose of M, denoted by M t c h M t = sp l l se l l sw v M t 4x2 is a “model by driving-condition” matrix, with 4 rows and 2 columns.
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If the rows and columns are interchanged, you get the transpose of M, denoted by M t c h M t = 17 23 sp l 22 30 l se l 17 24 l sw 16 19 v M t 4x2 is a “model by driving-condition” matrix, with 4 rows and 2 columns.
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The Environmental Protection Agency mandated in 5 years the fuel performance figures must increase 10%. This means every element in matrix M must be multiplied by 1.10, resulting in the matrix sp se sw v 1.1M = 18. 7 24.2 18.7 17.6 c 25.3 33 26.5 20.9 h This is called scalar multiplication, with 1.1 being called a scalar.
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EX: If A = 3 1 5 , find At, 2A, and -3A 4 0 -2 At=At= 3 4 | 1 0 5 -2 2A = 6 2 10 8 0 -4 -3A = -9 -3 -15 -12 0 6
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Two matrices with the same dimensions can be added or subtracted, by finding the sums or differences of the corresponding elements. EX: A = 381 40 -3 -215 B = 209 4 -6 -5 0 72 Find A + B and A – B. A + B = 5 8 10 8 -6 -8 -2 8 7 A – B = 1 8 -8 0 6 2 -2 -6 3
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EX: A = 2 -1 4 0 0 -8 B = -6 3 5 0 7 -4 Find A t + B and A + B t. A t = 2 4 0 -1 0 -8 B t = -6 0 3 7 5 -4 A t + B = -4 7 5 -1 7 -12 A + B t = -4 -1 7 7 5 -12
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Two matrices are equal if and only if they have the same dimensions and all corresponding elements (same row, same column) are equal. EX: Find the values of the variables for which the given statement is true. a b – 2 -3 = 7 2.5 c d 5 -1 -1 0 a b = 7 2.5 + 2 -3 c d -1 0 5 -1 a b = 9 -.5 c d 4 -1
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Solve the matrix equation for X 2 5 1 + 3X = 1 -4 3 4 3 -7 10 2 + 3X = 1 -4 6 8 3 -7 3X= 1 -4 – 10 2 = 3 -7 6 8 _1_ -9 -6 3 -3 -15 X = -3 -2 -1 -5
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