**Introduction:**

Applications of matrices are found in every branch of engineering, Including electromagnetism, quantum mechanics, quantum electrodynamics, classical mechanics, optics and computer graphics.

**Matrix:**

A set of mn numbers (real or complex) arranged in the form of a rectangular array of m rows and n columns is known as a matrix of order mxn. The matrices are generally denoted by capital letters such as A,B,C etc. The numbers in the matrix are called elements or entries of the matrix.

An mxn matrix is usually written as .

**Equality of Two Matrices**

Two matrices A = (a_{ij})_{mxn} and B = (B_{ij})_{pxq} are said to be equal if

i) they are of same order, i.e., m = p, n = q

and

ii) their corresponding elements are equal,

i.e., a_{ij} = b_{ij} for all i, j.

**Types of Matrices:**

**Real matrix: ** If all the elements of a matrix A are real numbers then the matrix A is called a real matrix.

**Complex matrix: **

If at least one of the elements of a matrix A is purely imaginary (or) complex then the matrix A is called complex matrix.

**Row matrix:**

If a matrix A has only one row and any number of columns then the matrix A is called a row matrix.

**Column matrix: **

If a matrix A has only one column and any number of rows then the matrix A is called a column matrix.

**Square matrix: **

If the number of rows is equal to the number of columns in a matrix then the matrix is called square matrix. A square matrix of order nxn is sometimes called as n-rowed matrix A or simply a square matrix of order ‘n’.

**Diagonal or principal diagonal elements:**

If A = (a_{ij})_{nxn} then the elements a_{ij} of a square matrix for which i = j,

i.e., the elements a_{11}, a_{22}, ……….… a_{nn} are called diagonal elements (or) leading diagonal elements.

a_{11}=1, a_{22}=5, a_{33}=8 are diagonal elements.

**Trace of a matrix:**

The sum of the diagonal elements of a square matrix is called trace of A and it is denoted by trace (A) or tr(A)

Thus, if A = (a_{ij})_{nxn}

Then tr(A_{nxn}) = a_{11}+a_{22}+……. +a_{nn} =

**Properties:**

If A and B are square matrices of order n then

(i) tr(AB) = tr(BA)

(ii) tr(AB) tr(A) tr(B)

(iii) tr(BA) tr(B) tr(A)

(iv) tr(A–B) = tr(A) – tr(B)

(v) tr(A+B) = tr(A)+tr(B)

(vi) tr(kA) = k tr(A) where k is a scalar.

**Rectangular matrix: **

If the number of rows is not equal to the number of columns in a matrix then the matrix is called a rectangular matrix.

**Diagonal matrix:**

If all the non-diagonal or off-diagonal elements in a square matrix are zero then the matrix is called diagonal matrix.

**Note:**

1. A matrix A = [a_{ij}]_{nxn} is diagonal matrix

if a_{ij} =

- If A and B are diagonal matrices then
- A+B is a diagonal matrix
- A–B is a diagonal matrix
- Adj(A), adj(B) are diagonal

matrices - A
^{–1}, B^{–1}are diagonal matrices

**Scalar matrix:**

If all the diagonal elements of a diagonal matrix are same or equal then the matrix is called a scalar matrix.

**Unit matrix or Identity matrix:**

If all the diagonal elements of a diagonal matrix are one then the matrix is called an identity matrix.

A unit matrix of order n is denoted by I_{n}.

**Upper triangular matrix:**

A = [a_{ij}]_{nxn} is an upper triangular matrix

**Triangular matrix:**

A matrix which is either upper triangular or lower triangular is called a triangular matrix.

**Strictly triangular matrix:**

A triangular matrix A = (a_{ij})_{nxn} is called a strictly triangular matrix

If a_{ii} = 0 for all i = 1, 2, …… n

**Idempotent matrix:**

If A^{2 }= A for a square matrix A of order n then A_{nxn} is called an idempotent matrix.

- If AB = A and BA = B then A and B are idempotent.
- If AB = BA = 0, then the sum of two idempotent matrices A and B is an

idempotent.

**Nilpotent matrix:**

If there exists a positive integer m for a square matrix A of order m such that A^{m} = 0 then the matrix A is called a nilpotent matrix.

If m is a least positive integer for which A^{m} = 0 then ‘m’ is called the index of the nilpotent matrix.

a nilpotent

matrix of index 2.

**Involuntary matrix:**

If A^{2} = I for a square matrix A then matrix A is called an involuntary.

**Transpose of a matrix:**

If a matrix B_{nxm} is obtained from a matrix A_{mxn} by changing its rows into columns and its columns into rows then the matrix B_{nxm} is called transpose of A and is denoted by A^{T} or A^{1}.

Thus, if A = (a_{ij})_{mxn} then A^{T} = [a_{ji}]_{nxm }

**Properties of Transpose:**

If A and B are two matrices and A^{T} and B^{T} are transpose of A and B respectively then

(i) (A^{T})^{T} = A

(ii) (AB)^{T} = B^{T}A^{T}

(iii) (kA)^{T} = kA^{T} where k is a scalar (real or

complex)

(iv) (AB)^{T} = A^{T}B^{T}

**Symmetric matrix:**

If a_{ij} = a_{ji }for all i, j in A = (a_{ij})_{nxn} then A_{nxn} is called a Symmetric matrix.

i.e. A^{T} = A

**Skew-symmetric matrix:**

A square matrix A is said to be Skew-symmetric matrix if A^{T} = –A.

A = (a_{ij})_{nxn} is a Skew-symmetric matrix

then a_{ij }= –a_{ji} i, j

**Properties of Symmetric & Skew-symmetric matrices:**

If A and A^{T} are any two square matrices then

- A + A
^{T}is symmetric - A – A
^{T}, A^{T}– A are skew-symmetric - AA
^{T}, A^{T}A are symmetric

If A and B are symmetric matrices then

(i) A B are symmetric

(ii) AB, BA are not symmetric

(iii) AB + BA is symmetric

(iv) AB – BA is skew-symmetric

If A and B are skew-symmetric then

(i) A B are skew-symmetric

(ii) AB, BA are not skew-symmetric

(iii) A^{2}, B^{2 }are symmetric

(iv) A^{2} B^{2} are symmetric

- Every square matrix A can be uniquely expressed as a sum of symmetric and Skew-symmetric matrices.
- If A and B are square symmetric matrices then AB is symmetric
- If A and B are skew-symmetric matrices then AB is symmetric A and B are commute