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Matrix Algebra

# Matrix Algebra - If If A and B are both m n matrices then...

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If A and B are both m × n matrices then the sum of A and B , denoted A + B , is a matrix obtained by adding corresponding elements of A and B . - - = 3 1 0 2 2 1 A - - = 4 1 2 4 0 3 B - = + 2 B A elements of A and B . add these - - = 3 1 0 2 2 1 A - - = 4 1 2 4 0 3 B - - = + 2 2 B A add these - - = 3 1 0 2 2 1 A - - = 4 1 2 4 0 3 B - - = + 6 2 2 B A add these - - = 4 1 2 4 0 3 B - - = + 2 6 2 2 B A add these - - = 3 1 0 2 2 1 A - - = 4 1 2 4 0 3 B - - = + 0 2 6 2 2 B A add these - - = 3 1 0 2 2 1 A - - = 4 1 2 4 0 3 B - - - = + 1 0 2 6 2 2 B A add these
A B B A + = + C B A C B A + + = + + ) ( ) (

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If A is an m × n matrix and s is a scalar, then we let kA denote the matrix obtained by multiplying every element of A by k. This procedure is called scalar multiplication. ( 29 ( 29 ( 29 ( 29 k hA kh A k h A kA hA k A B kA kB = + = + + = + - - = 3 1 0 2 2 1 A ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 - - = - - = 9 3 0 6 6 3 3 3 1 3 0 3 2 3 2 3 1 3 3 A PROPERTIES OF SCALAR MULTIPLICATION
The m × n zero matrix , denoted 0 , is the m × n matrix whose elements are all zeros. ( 29 0 0 0 ) ( 0 = = - + = + A A A A A 0 0 0 0 [ ] 0 0 0 2 × 2 1 × 3

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The multiplication of matrices is easier shown than put into words. You multiply the rows of the first matrix with the columns of the second adding products - - = 1 4 0 1 2 3 A - - = 1 3 3 1 4 2 B Find AB First we multiply across the first row and down the first column adding products. We put the answer in the first row, first column of the answer. ( 29 2 3 ( 29 ( 29 1 2 - - + ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 5 3 1 1 2 2 3 = - + - - +
- - = 1 4 0 1 2 3 A - - = 1 3 3 1 4 2 B Find AB We multiplied across first row and down first column so we put the answer in the first row, first column.

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