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For example:
14
12 3
‘
A = A OR A = 24
T
45 6 ×23 36
×32
Multiplication of a Matrix by a Scalar
A scalar is any number. So, if A is a matrix and k is a scalar, then kA is another matrix which is obtained by multiplying
each element of A by the scalar k. For example:
15 3
A = k3
=
2 47
1 3 53 33 × 3 15 9
×
×
=
kA 3A = ⇒
×
×
×
2 3 4 3 7 3 6 12 21
Multiplication of Matrices
The product of two matrices A and B is defined if the number of columns of A is equal to the number of rows of B.
Consider A = [a ], an m × n matrix and B = [b ] is an n × p matrix. Then the product of the matrices A and B is the
jk
ij
th
th
matrix C of order m × p. To get the (i, k)th element i.e c of the matrix C, we take the i row of A and k column of B,
ik
multiply them elementwise and take the sum of all these products. For example:
5 1 26 0
A = B =
2 3 2 2 13 4 23
×
×
equal
AB =
23
×
Resultant Matrix, AB is of
the order 2 x 3
Take the first row of matrix A and first column of matrix B. Multiple the corresponding elements and then add them
as shown. Repeat this step for all columns of B.
Step 1: 51 26 0
A = B =
23 13 4
11 ? ?
⇒ (5 2 1 1) ⇒ 11 ⇒
+×
×
? ??
Resultant Matrix, AB getting filled
Step 2: 51 26 0
A = B =
23 13 4
11 33 ?
+×
⇒ 5 6 1 3 ⇒ 33 ⇒
×
? ? ?
Step 3: A = 51 B = 26 0
23 13 4
11 33 4
⇒ 5 0 14 ⇒ ⇒
+×
×
4
? ? ?
164 Touchpad Artificial Intelligence (Ver. 2.0)-XI

