# Symmetric Matrix

A square matrix in which

*a*_{ij}= a_{ji}

**Example :**

A =

For symmetric matrix, A

^{T}= A

# Skew Symmetric Matrix

A square matrix in which

*a*_{ij}= â€“ a_{ji}All principal diagonal elements in skew symmetric matrix are zero.

**Example :**

A =

In skew symmetric matrix, A

^{T}= â€“ A

# Hermitian Matrix

A square matrix in which *a _{ij}* is the conjugate complex of

*a*

_{ji}*i.e. *

**Example :**

*a*_{12} = 2 + 3*i*, *a*_{21} conjugate complex of *a*_{12} = 2 â€“ 3*i*

# Skew Hermitian Matrix

A square matrix in which

*a*=_{ij}

**Note : **Principal diagonal elements of skew hermitian matrix should be zero or pure imaginary numbers.

# Singular Matrix

*A square matrix is a singular matrix if its determinant is zero. Otherwise, it is a non-singular matrix.*

# Orthogonal Matrix

A square matrix A is called an orthogonal matrix if AA

^{T}= A^{T}A = I

# Trace Of A Matrix

Let = [

*a*_{ij}]_{nÃ—n}be a square matrix. Then the sum of all diagonal elements of A is called the trace of A and is denoted by tr (A). Thus, tr(A) =

# Idempotent Matrix

*A square matrix A is called an idempotent matrix if A*

^{2}=

*A*.

**Properties of transpose of a matrix**

- (A + B)
^{T}= A^{T}+ B^{T} - (
*p*A)^{T}=*p*A^{T} - (AB)
^{T}= B^{T}A^{T}(not A^{T}B^{T}) - AA
^{T}= A^{T}A

**Inverse of a Matrix**

A

^{â€“1}=**Properties of Inverse of Matrix**

- (A
^{â€“1})^{â€“1}= A - (AB)
^{â€“1}= B^{â€“1}A^{â€“1} - (KA)
^{â€“1}= K^{â€“1}A^{â€“1}= - | A
^{â€“1}| = - adj. (AB)
^{ }= (adj. B) (adj. A) - | adj. A| = | A |
^{n}^{â€“1}where A is square matrix.

# Echelon Matrix

**A matrix is said to be echelon matrix if :**

- Leading entry of each non zero row is 1
- The number of zeros proceeding this entry 1 is larger than number of proceeding rows.

**Example :**

A =

Consistency of a Algebraic Equations

Note :

Note :

An equation is said to be homogenous if

*b*= 0 i.e. of form AX = 0.A system of equations is said to be consistent, if there are one or more solution for the equation.

A system of equations is said to be inconsistent, if there is no solution. When we reduce a matrix A to echelon form it is called argument matrix A*.

If

*p*(A*) =*p*(A) the equations are consistentIf

*p*(A*) â‰*p*(A) the equations are inconsistent.In a homogeneous system of equation (AX = 0),

If

*p*(A) = no. of unknowns (*n*), it has no solutionIf

*p*(A) < no of unknown (*n*), it has infinite solutions.Consider an augmented matrix AB which is reduced to echelon form.

If

*p*(A) â‰*p*(AB), system is inconsistentIf

*p*(A) =*p*(AB) = no. of unknown (*n*), system is consistent with unique solution.If

*p*(A) =*p*(AB) < number of unknowns (n), system is consistent with infinite solutions.**Eigen Values and Eigen Vectors**

For an

*n Ã— n*matrix A and a scalar Î», characteristic matrix of A isA â€“ Î»I. The equations | A â€“ Î»I | = 0 is called characteristic equation and its roots is called Eigen values.

For Eigen values (Î») of matrix A, there is a non-zero vector X such that AX = Î»X is called characteristic vector (Eigen vector) of A.

**Relations between Eigen values and Eigen vector**

- Î» is a Eigen values of a matrix A if and only if there exists a non-zero vector X such that
- AX = Î» X
- If X is a characteristic vector of a matrix A corresponding to the characteristic value Î», then KX is also a characteristic vector of A corresponding to the same characteristic value Î». Here K is any non-zero scalar.
- If X is a Eigen vector of a matrix A, then X cannot correspond to more than one Eigen value of A.
- Eigen vectors corresponding to distinct characteristic roots of a matrix are linearly independent.

**Properties of Eigen Values**

- Sum of eigen values is equal to sum of element of principal diagonal of the matrix i.e.
- Product of eigen values = determinant of the matrix.
- If Î» is eigen value of an orthogonal matrix, A then is also its eigen value.
- If Î» is eigen value of matrix A, then is eigen value of matrix A
^{â€“1}. - If Î»
_{1}, Î»_{2}, Î»_{3}... are eigen values of matrix A, then Î»_{1}, Î»^{n}_{2}, Î»^{n}_{3}... are eigen values of matrix A^{n}.^{n} - Both A and A
^{T}will have same eigen values. - Eigen values of triangular matrix is equal to its diagonal elements.
- Eigen values of real symmetric matrix are all real.
- Eigen values of skew hermitian matrix are purely imaginary or zero.
- It Î» is eigen value of A, then is eigen value of adj. (A).