1. Let A = (a_{ij}) be an m × n matrix:

2. The transpose of A is the n × m matrix

The rows of A are the columns of A^{t}. For example, if

then

Note that (A^{t})^{t} = A.

3. If c is a scalar (i.e. a real or complex number) and A = (a_{ij} )is an m × n
matrix, then cA is the m × n

matrix with entries ca_{ij} . So if A is given by (1) and c = 2, then

4. If A = (a_{ij}) and B = (b_{ij}) are m × n matrices, then the sum A + B is the m
× n matrix with entries

a_{ij} + b_{ij} . So if A is given by (1) and

then

5. If A is an m × n matrix and B is n × k, then the product AB is the m × k
matrix whose ijth entry is

the dot product of the ith row of A and the jth column of B. For example, let

Since A is 2 × 3 and B 3 × 3, the product AB will be a 2 × 3 matrix. In particular,

6. The m×m identity matrix I_{m} is the m×m matrix with main diagonal entries 1
and 0’s everywhere else.

So, for example,

7. An m × m matrix A is invertible, or nonsingular if there is an m × m
matrix A^{−1} such that

The matrix A^{−1} is called the inverse of A.

8. The determinant of the 2 × 2 matrix

is

If det(A) ≠ 0, then A is invertible, and its inverse is

Since the matrix

has det(A) = 6, it is invertible and