Product of Matrices

Product of Matrices,

Let , and ,

for matrix A of order m x n,and matrix B of order p x q, product of matrices exist if and only if . Then the product of matrices is where the order of matrix D is x by y and , . The elements of D are

Or

For example,

From the definition of matrix multiplication, it is possible to have

without or and

without or .

Properties of Matrix Multiplication

If both matrix B premultiplied by matrix A, and matrix B postmultipled by matrix A, are defined, . In general, matrix multiplication is not commutative.

. Multiplication of matrices is associative.

For matrix A of order m x n, matrix B of order p x q and matrix C of order s x t, product of matrices exist if and only if and . Then element of is

And the element of is

Rearrange the order of terms

And equals to the element of

or .

Matrix multiplication is distributive with respect to addition.

If , then

and .

If A and D are of same order and B and C are square matrices, then A, B, C and D are all square matrices.

Since equals and the element is

And equals to the element of .

Identity Matrix

The identity under matrix multiplication is the identity matrix, that is

Let , and then

and

In order to have the same order as A, for element x, order of k should equal to i and for element y, order of l should equal to j, and therefore the identity matrix must be square.

According to the equality of matrices,

only when , and only when .

The form of identity matrix is

.

If , that is , then A must be a square matrix.

And therefore it is also called unit matrix.

Inverse Matrix

If A has an multiplication inverse and then A must be square.

If a matrix has an inverse, the inverse is unique.

Assume both B and C are inverses of A, then and .

Premultiply first equation by C, imply .

From second equation, imply and the inverse is unique.

Division of matrices has not been defined, the multiplying of the reciprocal of a matrix can be replaced by multiplying the inverse of the matrix. Therefore solving B in can be obtained by premultiplying .

Let , If both matrices have an inverse, then . That is

The inverse of a matrix's inverse equals to the matrix itself, . Let then