In details

Matrix multiplication


The product of one matrix by another is not determined by the product of its respective elements.

Thus, the product of matrices A = (aij)m x p and B = (bij)p x n is the matrix C = (cij) m x nwhere each element çijis obtained by summing the products of the corresponding elements of the ith row of A by the elements of the ith column B.

Let's multiply the matrices to understand how to get each element çij:

  • 1st row and 1st column

  • 1st row and 2nd column

  • 2nd row and 1st column

  • 2nd row and 2nd column

Like this, .

Now look what would happen if you did the opposite, ie multiply B by A:

Therefore, .A, ie for matrix multiplication is not worth the commutative property.

Let's look at another example with the matrices :


From the definition, we have the product matrix A. B only exists if the number of columns of THE equals the number of lines of B:

The product matrix will have the number of rows of Y and the number of columns of B (n):

  • If A3 x 2 and B 2 x 5 then (A. B) 3x5

  • If A 4x1 and B 2x3, then there is no product

  • If A 4 x 2 and B 2x1then (A. B) 4x1

Properties

Once the conditions of existence for matrix multiplication are verified, the following properties apply:

a) associative: (A. B). C = A. (B. C)

b) distributive in relation to addition:. (B + C) = A. B + A. C or (A + B). C = A. C + B. Ç

c) neutral element:. Ino = Ino . A = A, where Ino the order identity matrix no

We have seen that commutative property generally does not apply to matrix multiplication. Also not worth the cancellation of the product, ie: being 0 m x n a null matrix, A .B = 0m x n does not necessarily imply that A = 0m x n or B = 0m x n.

Inverse matrix

Given an array THEsquare order no, if there is an array THE'of the same order as. A '= A'. A = Ino , then THE' is inverse matrix of THE . We represent the inverse matrix by THE-1 .

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