Machinery's Handbook, 31st Edition
124 Matrix Operations Matrix Operations Matrix Addition and Subtraction.— The sum or difference of matrices is determined simply by adding or subtracting the corresponding elements of each matrix. So, matrix C is the result of adding or subtracting matrices A and B ; the entries are com- bined as follows: a ij + b ij = c ij or a ij - b ij = c ij Matrices must be the same size, m × n , to be combined this way. That is, A mn + B mn = C mn ; A mn - B mn = C mn . An efficient way to indicate both + and - is with the symbol ± . Thus, in the matrix display below, a ij ± b ij covers both matrix operations. Example 1:
4 6 –5 5 –7 8 –8 6 –7
8 –2 6 –6 9 5 9 –2 2
4+8 ( ) 6– 2 ( ) –5 +6 ( ) 5– 6 ( ) –7 +9 ( ) 8+5 ( ) –8 +9 ( ) 6– 2 ( ) –7 +2 ( )
12 4 1 –1 2 13 1 4 –5
=
=
+
Matrix Multiplication.— Two matrices can be multiplied only if the number of columns in the first matrix equals the number of rows of the second matrix. For example, a 1 × 3 matrix can multiplied by a 3 × 2 matrix, but not the other way around. Or a 2 × 4 by a 4 × 3, in that order. In general, an m × n and an n × p can be multiplied; the product matrix is an m × p matrix. Matrix multiplication is not commutative, that is, A × B is not necessarily equal to B × A . The steps in matrix multiplication are shown in the instructive example below for a gen- eral 3 × 2 matrix multiplied by a 2 × 1 matrix. The result is a 3 × 1 matrix. 1 4 1 · 0 + 4 ∙ –2 –8
0
–3
1
–3 ∙ 0 + 1 ∙ –2
–2
×
=
=
–2
2 –1
2 ∙ 0 + –1 ∙ –2
2
In general, each entry ( AB ) ij in the product matrix (where i is the row number and j is the column number) is equal to a i 1 b 1 j + a i 2 b 2 j + … + a in b nj . For example:
a 11 b 11 + a 12 b 21
a 11 b 12 + a 12 b 22
a 11 a 12 a 21 a 22 a 31 a 32
b 11 b 12 b 21 b 22
a 21 b 11 + a 22 b 21
a 21 b 12 + a 22 b 22
×
=
a 31 b 11 + a 32 b 21
a 31 b 12 + a 32 b 22
3 × 2 matrix
2 × 2 matrix
3 × 2 matrix
Example 2: 1 2 3 4 5 6 3 2 1 ×
7 8 9 1 2 3 4 5 7
1 7 ⋅ 2 1 ⋅ 3 4 ⋅ + + ( ) 1 8 ⋅ 2 2 ⋅ 3 5 ⋅ + + ( ) 1 9 ⋅ 2 3 ⋅ 3 7 ⋅ + + ( ) 4 7 ⋅ 5 1 ⋅ 6 4 ⋅ + + ( ) 4 8 ⋅ 5 2 ⋅ 6 5 ⋅ + + ( ) 4 9 ⋅ 5 3 ⋅ 6 7 ⋅ + + ( ) 3 7 ⋅ 2 1 ⋅ 1 4 ⋅ + + ( ) 3 8 ⋅ 2 2 ⋅ 1 5 ⋅ + + ( ) 3 9 ⋅ 2 3 ⋅ 1 7 ⋅ + + ( ) = 7+2+12 ( ) 8+4+15 ( ) 9+6+21 ( ) 28+5+24 ( ) 32+10+30 ( ) 36+15+42 ( ) 21+2+4 ( ) 24+4+5 ( ) 27+6+7 ( ) = 21 27 36 57 72 93 27 33 40 =
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