A matrix represents a collection of numbers arranged in an order of rows and columns. It is necessary to enclose the elements of a matrix in parentheses or brackets.

A matrix with 9 elements is shown below.

This Matrix [M] has 3 rows and 3 columns. Each element of matrix [M] can be referred to by its row and column number. For example, a_{23}=6

**Order of a Matrix :**

The order of a matrix is defined in terms of its number of rows and columns.

Order of a matrix = No. of rows ×No. of columns

Therefore Matrix [M] is a matrix of order 3 × 3.

**Transpose of a Matrix :**

The transpose [M]^{T} of an m x n matrix [M] is the n x m matrix obtained by interchanging the rows and columns of [M].

if A= [a_{ij}] mxn , then A^{T} = [b_{ij}] nxm where b_{ij} = a_{ji}

**Properties of transpose of a matrix:**

- (A
^{T})^{T}T = A - (A+B)
^{T}T = A^{T}T + B^{T}T - (AB)
^{T}T = B^{T}TA^{T}T

**Singular and Nonsingular Matrix:**

- Singular Matrix: A square matrix is said to be singular matrix if its determinant is zero i.e. |A|=0
- Nonsingular Matrix: A square matrix is said to be non-singular matrix if its determinant is non-zero.

**Properties of Matrix addition and multiplication:**

- A+B = B+A (Commutative)
- (A+B)+C = A+ (B+C) (Associative)
- AB ≠ BA (Not Commutative)
- (AB) C = A (BC) (Associative)
- A (B+C) = AB+AC (Distributive)

**Square Matrix:** A square Matrix has as many rows as it has columns. i.e. no of rows = no of columns.

**Symmetric matrix:** A square matrix is said to be symmetric if the transpose of original matrix is equal to its original matrix. i.e. (A^{T}) = A.

**Diagonal Matrix:** A Symmetric matrix is said to be diagonal matrix where all the off diagonal elements are 0.

**Identity Matrix:** A diagonal matrix with 1s and only 1s on the diagonal. Identity matrix is denoted as I.

**Orthogonal Matrix:** A matrix is said to be orthogonal if AA^{T} = A^{T}A = I

**Idemponent Matrix:** A matrix is said to be idemponent if A^{2} = A

**Involutary Matrix:** A matrix is said to be Involutary if A^{2} = I.

Note: Every Square Matrix can uniquely be expressed as the sum of a symmetrix matrix and skew symmetric matrix. A = 1/2 (AT + A) + 1/2 (A – AT).

**Adjoint of a square matrix:**

**Properties of Adjoint:**

- A(Adj A) = (Adj A) A = |A| I
_{n} - Adj(AB) = (Adj B).(Adj A)

Inverse of a square matrix:

A^{-1} = Adj A / |A| ; |A|#0

**Properties of inverse:**

1. (A^{-1})^{-1} = A

2. (AB)^{-1} = B^{-1}A^{-1}

3. only a non singular square matrix can have an inverse.

**Where should we use inverse matrix?**

If you have a set of simultaneous equations:

7x + 2y + z = 21

3y – z = 5

-3x + 4y – 2x = -1

As we know when AX = B, then X = A^{-1}B so we calculate inverse of A and by multiplying it B, we can get the values of x, y and z.

**Trace of a matrix:** trace of a matrix is denoted as tr(A) which is used only for square matrix and equals the sum of the diagonal elements of the matrix. For example:

This article is contributed by **Nitika Bansal**. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.