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Unformatted text preview: Matrices
matrix product Matrix multiplication A · B is deﬁned only if the number of columns of matrix A is the same as the number of rows of matrix B. Amxp · Bpxn = Cmxn , p is the number of columns of A and p is the number of rows of B. The resulting matrix C has the same number of rows as A and the same number of columns as B Matrix multiplication is deﬁned in terms of multiplying a row matrix by a column matrix. Both matrices must have the same number of elements. A= A·B= 1 2 · 3 4 = 1 2 and , B = = 3 4 11 (1) (2) 1·3+2·4 Matrices
matrix product C=A·B= 1 3 2 4 · 4 2 3 1 = c11 c21 c12 c22 = 8 20 5 13 (3) cij is the element in row i and column j of matrix C = AB cij is found by multiplying row i of matrix A by column j of matrix B jth ith row of A · column cij = of B c11 = c21 = 1 3 2 4 · · 4 2 4 2 = 8, c12 = = 20, c22 = 1 3 2 4 · · 3 1 3 1 =5 = 13 (4) (5) Matrices
matrix product: example Write the system of equations in matrix form. x + 2y = 5 3x + 4y = 19 x + 2y 3x + 4y 1 3 2 4 1 3 x y =
2x1 (6) (7) 5 11 5 11 5 11 (8)
2x1 (9) ·
2x2 =
2x1 (10)
2x1 (11) 2 4 · x y = (12) (13) A·X=B (14) Matrices
AB = BA Three cases: AB is deﬁned but BA is not deﬁned. AB and BA are both deﬁned but not the same AB does equal BA Summary: In general AB = BA but in some cases it is true that AB = BA ...
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This note was uploaded on 04/04/2010 for the course MATH 160 taught by Professor Doyle during the Spring '08 term at Ill. Chicago.
 Spring '08
 DOYLE
 Math, Multiplication, Matrices

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