# Symmetric Matrix

A square matrix in which aij = aji

Example :

A =
For symmetric matrix, AT = A

# Skew Symmetric Matrix

A square matrix in which aij = â€“ aji
All principal diagonal elements in skew symmetric matrix are zero.

Example :
A =

In skew symmetric matrix, AT = â€“ A

# Hermitian Matrix

A square matrix in which aij is the conjugate complex of aji

i.e.

Example :

a12 = 2 + 3ia21 conjugate complex of a12 = 2 â€“ 3i

# Skew Hermitian Matrix

A square matrix in which aij =

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 AAT = ATA = I

# Trace Of A Matrix

Let = [aij]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 A2 = A.

Properties of transpose of a matrix

• (A + B)T = AT + BT
• (pA)T = pAT
• (AB)T = BTAT (not ATBT)
• AAT = ATA
Inverse of a Matrix
Aâ€“1 =

Properties of Inverse of Matrix
• (Aâ€“1)â€“1 = A
• (AB)â€“1 = Bâ€“1Aâ€“1
• (KA)â€“1 = Kâ€“1Aâ€“1 =
• | Aâ€“1 | =
• | adj. A| = | A |nâ€“1 where A is square matrix.

# Echelon Matrix

A matrix is said to be echelon matrix if :
1. Leading entry of each non zero row is 1
2. The number of zeros proceeding this entry 1 is larger than number of proceeding rows.

Example :
A =

Consistency of a Algebraic Equations

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 consistent
If 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 solution
If 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 inconsistent
If 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 is
A 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 Aâ€“1.
• If Î»1, Î»2, Î»3... are eigen values of matrix A, then Î»1n, Î»2n, Î»3n ... are eigen values of matrix An.
• Both A and AT 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).