### Matrix

Matrix refer to a rectangular formation (array or table) of certain number(real or complex) or of certain function.

Matrics are represented bt the capital letters A, B, C, etc.

### Order of matrix

A matrix having m rows and n columns and enclosed by a square bracket [ ] is called mxn matrix (read “m by n matrix”).

An mxn matrix is expressed as

or A = (a_{ij})_{m x n}

**For Example:** The order or dimension of a matrix is the ordered pair having as a first component the number of rows and as a second component the number of columns in the matrix. If there are 3 rows and 2 columns in a matrix, then its order is written as (3, 2) or (3 x 2) read as three by two. In general, if m is rows and n are columns of a matrix, then its order is (m x n).

The m,n numbers are known as **Element** of the matrix. They are accompanied by two indices the first of which indicated the row on which it appears and second the appropriate column.

are matrices of orders (2 x 3), (3 x 1) and (4 x 4) respectively.

### Types of Matrices

#### Row Matrix:

A matrix of order 1 x n is called a row matrix or a row vector. Its consisting of a single row is called a **row matrix** or a **row vector**.

A = [1, 2, 3]

#### Column Matrix:

A matrix of order m x 1 is called a **column matrix** or a **column vector**. Its consisting of a single column.

#### Null or Zero Matrix:

A matrix in which each element is “0” is called a Null or Zero matrix. Zero matrices are generally denoted by the symbol O. This distinguishes zero matrix from the real number 0.

is a zero matrix of order 2 x 4.

The matrix O_{m x n} has the property that for every matrix A_{m x n},

A + O = O + A = A

#### Square matrix:

A matrix having same numbers of rows and columns is called a **square matrix**. A matrix A of order m x n can be written as A_{m x n}. If m = n, then the matrix is said to be a square matrix. A square matrix of order n x n, is simply written as A_{n}.

are square matrix of order 2 and 3

### Main or Principal (leading)Diagonal

The principal diagonal of a square matrix is the ordered set of elements a_{ij}, where i = j, extending from the upper left-hand corner to the lower right-hand corner of the matrix. Thus, the principal diagonal contains elements a_{11}, a_{22}, a_{33} etc.

For example, the principal diagonal of

consists of elements 1, 2 and 0, in that order.

### Cases of a square matrix:

#### Diagonal matrix

A square matrix in which all elements are zero except those in the main or principal diagonal is called a diagonal matrix. Some elements of the principal diagonal may be zero but not all.

**For Example:**

are diagonal matrices.

In general

is a diagonal matrix if and only if

a_{ij} = 0 for i ≠ j

a_{ij} ≠ 0 for at least one i = j

#### Scalar matrix

A diagonal matrix in which all the diagonal elements are same, is called a scalar matrix.

#### Identity Matrix or Unit matrix

A scalar matrix in which each diagonal element is 1(unity) is called a **unit matrix**. An identity matrix of order n is denoted by I_{n}.

are the identity matrices of order 2 and 3.

In general,

is an identity matrix if and only if

a_{ij} = 0 for i ≠ j and a_{ij} = 1 for i = j

**Note:** If a matrix A and identity matrix I are comformable for multiplication, then I has the property that AI = IA = A i.e., I is the identity matrix for multiplication.

#### Equal Matrices

Two matrices A and B are said to be equal if and only if they have the same order and each element of matrix A is equal to the corresponding element of matrix B i.e for each i, j, a_{ij} = b_{ij}

then A = B because the order of matrices A and B is same and a_{ij} = b_{ij} for every i, j.

#### Upper Triangular Matrix

If in a matrix, elements for which i > j are zeros, then it is known as **upper triangular matrix**.

#### Lower Triangular Matrix

If in a matrix, elements for which i < j are zeros, then it is known as **lower triangular matrix**.

#### The Negative of a Matrix:

The negative of the matrix A_{m x n}, denoted by –A_{m x n}, is the matrix formed by replacing each element in the matrix A_{m x n} with its additive inverse.

** For example**,

for every matrix A_{m x n}, the matrix –A_{m x n} has the property that

A + (–A) = (–A) + A = 0

i.e., (–A) is the additive inverse of A.

The sum B_{m – n} + (–A_{m x n}) is called the difference of B_{mxn} and A_{mxn}

and is denoted by B_{m x n} – A_{m x n}.

### Operations on matrices

#### Addition of Matrix:

Two matrices can be added only if they are of the same order. In such case, the respective elements of the two matrices are added e.g

if A = |a_{ij}|_{m x n} , B = |b_{ij}|_{m x n}

then A + B = |a_{ij} + b_{ij}|_{m x n}

#### Properties of Matrix addtion

• It is commutative i.e. A + B = B + A

• It is associative i.e. ( A + B ) + C = A + ( B + C )

• It is odditive i.e. A + 0 = 0 + A = A

• Existance of additive inverse i.e. A + (-A) = 0 = (-A) + A

#### Subtraction of Matrix:

Two matrices can be subtract only if they are of the same order. In such case, the respective elements of the two matrices are subtracted e.g

if A = |a_{ij}|_{m x n} , B = |b_{ij}|_{m x n}

then A – B = |a_{ij} – b_{ij}|_{m x n}

#### Multiplication of Matrix:

Matrix multiplication is possible only if the number of columns of the first matrix is equal to the number of rows of the second matrix.

**For Example:**

#### Properties of Matrix Multiplication

• It is associative i.e. ( AB )C = A( BC )

• It is distributive i.e. A( B + C ) = AB + AC

or (A + B)C = AC +BC

**Note :**

1 . Multiplication of matrices is not commutative i.e., AB ≠ BA in general.

2 . For matrices A and B if AB = BA then A and B commute to each other

3 . A matrix A can be multiplied by itself if and only if it is a square matrix.The product A.A in such cases is written as A2. Similarly we may define higher powers of a square matrix i.e.,

A . A^{2} = A^{3} , A^{2}. A^{2} = A^{4}

4. In the product AB, A is said to be pre multiple of B and B is said to be post multiple of A.

#### Multiplication of a scalar to a matrix

If A = (a_{ij})_{m x n} and k is a scalar, then

k.A = (k a_{ij})_{m x n}

If we multiply a matrix by some scalr, the all the element of the matrix are multiplied by scalar.

I hope, this article will help you a lot to understand the **Matrix Part-1 **. If you still have any doubts and problems with any topic of mathematics you can ask your problem in the Ask Question section. You will get a reply shortly.