Lesson 12.2 Matrix Multiplication. 3 Row and Column Order The rows in a matrix are usually indexed 1...
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Transcript of Lesson 12.2 Matrix Multiplication. 3 Row and Column Order The rows in a matrix are usually indexed 1...
3
Row and Column Order
The rows in a matrix are usually indexed 1 to m from top to bottom. The columns are usually indexed 1 to n from left to right. Elements are
indexed by row, then column.
nmmm
n
n
ji
aaa
aaa
aaa
a
,2,1,
,22,21,2
,12,11,1
, ][
A
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.
310
221A
930
663
331303
2323133A
In matrix algebra, a real number is often called a SCALAR. To multiply a matrix by a scalar, you multiply each entry in the matrix by that scalar.
14
024
Multiplying Matrices by a scalar
)1(4)4(4
)0(4)2(4
416
08
Scalar Multiplication - each element in a matrix is multiplied by a constant.
1. 2 7 1 0 -14 2 0
2. 4 2 0
4 5
-8 0
16 -20
3. 5
1 2 3
4 5 6
7 8 9
-5 -10 15
-20 25 -30
35 -40 45
86
54
30
212
)8(360
52412
-2
6
-3 3
-2(-3)
-5
-2(6) -2(-5)
-2(3) 6 -6
-12 10
Example
PEMDAS – parenthesis first, do the matrix addition
Do the scalar multiplication
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
140
123A
13
31
42
B
Find AB
Multiply across the first row and down the first column adding products. Put the answer in the first row, first
column of the answer matrix.
23 1223 5311223
140
123A
13
31
42
B
Find AB
We multiplied across first row and down first column so we put the answer in the first row, first column.
5AB
Now we multiply across the first row and down the second column and we’ll put the answer in the first row, second column.
43 3243 7113243
75AB
Now we multiply across the second row and down the first column and we’ll put the answer in the second row, first column.
20 1420 1311420
1
75AB
Now we multiply across the second row and down the second column and we’ll put the answer in the second row,
second column.
40 3440 11113440
111
75AB
Notice the sizes of A and B and the size of the product AB.
Con’t
• You can multiply two matrices A and B only if the number of columns of A is equal to the number of rows of B.
Examples:
1. 2 1
3 4
3 9 2
5 7 6
2(3) + -1(5) 2(-9) + -1(7) 2(2) + -1(-6)
3(3) + 4(5) 3(-9) + 4(7) 3(2) + 4(-6)
1 25 10
29 1 18
3 9 2 2 1.
5 7 6 3 4
2
Dimensions: 2 x 3 2 x 2
*They don’t match so can’t be multiplied together.*
1 2 1 1
1 3 2 7
2 6 1 8
x
y
z
3.
x 2y z 1
x 3y 2z 7
2x 6y z 8