Basic Operations

Matrix

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Addition  The sum A+B of two m-by-n matrices A and B is calculated entrywise:

(A + B)_i,j = A_i,j + B_i,j, where 1 ≤ i ≤ m and 1 ≤ j ≤ n.

  

Scalar Multiplication

The scalar multiplication cA of a matrix A and a number c (also called a scalar in the parlance of abstract algebra) is given by multiplying every entry of A by c:

(cA)_i,j = c · A_i,j.

Transpose

The transpose of an m-by-n matrix A is the n-by-m matrix A^T (also denoted A^tr or ^tA) formed by turning rows into columns and vice versa:

(A^T)_i,j = A_j,i.

MATRIX MULTIPLICATION 

Multiplication of two matrices is defined only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot-product of the corresponding row of A and the corresponding column of B: