Question

Explain how the rank of a matrix and existence and uniqueness of solutions of “Systems of Linear Equations” are related

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Explain how Eigenfunctions”, “Eigenvalues” and “Orthogonality” terms and concepts are defined in Matrix algebra.

Answer 1

# Let A be any matrix of order nm.

The rank of a matrix is the number of non-zero rows in row echelon form of matrix.

By writing system of linear equations in matrix form, let [ A | D ] denotes augmented matrix.

The solution exists & it is unique if & only if Rank (A ) = Rank ( A| D ).

# Let A be any square matrix of order n.

Eigenvalue :The eigenvalue is defined to be root of characteristics polynomial of A .

i.e. To find eigenvalue we will solve determinant ( A - I ) = 0 , then we will get characteristics equation & its roots are called as eigenvalues of A.

Eigenfunctions :To find eigenfunctions corresponding to eigenvalue , we have to solve the system (A - I ) X = 0.

where I is identity matrix of order n. The solution of this system are eigenfunctions corresponding to eigenvalue .

Orthogonality : A matrix "A" is said to be orthogonal if A.A^t = A^t.A = I . (A^t denotes transpose of A)