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Echelon Matrix


A matrix is said to be echelon marix if :

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

Ex. A = 598.png

Consistency of a Algebraic Equations

Note :

An equation is said to be homogeneus 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*) ≠ 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) ≠ 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 × n matrix A and a scalar λ, characteristic matrix of A is
A – λI. The equations | A – λI | = 0 is called characterisic equation and its roots is called eigen values.

For eigen values (λ) of matrix A, there is a non zero vector X such that AX = λX is called characteristic vector (eigen vector) of A.

Relations between Eigen values and Eigen vector

  • λ is a eigen values of a matrix A if and only if there exists a non-zero vector X such that
  • AX = λ X
  • If X is a characteristic vector of a matrix A corresponding to the characteristic value λ, then KX is also a characteristic vector of A corresponding to the same characteristic value λ. 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 λ is eigen value of an orthogonal matrix, A then 618.png is also its eigen value.
  • If λ is eigen value of matrix A, then 628.png is eigen value of matrix A–1.
  • If λ1, λ2, λ3... are eigen values of matrix A, then λ1n, λ2n, λ3n ... 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 symmetrix matrix are all real.
  • Eigen values of skew hermitian matrix are purely imaginary or zero.
  • It λ is eigen value of A, then 664.png is eigen value of adj. (A).

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