Multiplication of matrices

Two matrices A and B are conformable for multiplication if the number of columns in A is same as the number of rows in B.

A is premultiplier matrix and B is called post multiplier matrix.

AB is defined in the following way.

(AB) _ij = Σ(r= 1 to n) a_irb_rj
= a_i1b_1j+a_i2b_2j+…a_imb_mj

When AB exists, BA may or may not exist.

Properties of matrix multiplication

1. Matrix multiplication is not commutative in general.
2. Matrix multiplication is associative.
3. Matrix multiplication is distributive over matrix.
4. If A is an m x n matrix, then I_mA = A = AI_n
5. The product of two matrices can be the null matrix while neither of them is the null matrix.
6. Product of the matrix with a null matrix is always a null matrix.
7. If AB = 0 when either of them is not zero, it does not imply BA is zero.


Positive integral powers of A
Aⁿ⁺¹ = AⁿA
(A^m)ⁿ = (A^-)mn


Matrix polynomial = a₀Aⁿ+a₁A^n-1+…+a_n-1A+a_nI_n.