GATE Mathematics : Linear Algebra

ONLINE TEST : GATE MATHS :- LINEAR ALGEBRA

Important Types Of Matrices :

1) Null/ Zero matrix : all elements are zero

2) Diagonal Matrix : square matrix whose all non-diagonal elements are zero.

3) Scalar matrix : diagonal matrix whose all diagonal elements are
equal to zero.

4) Symmetric matrix : In a Square matrix for all i and j , a_ij = a_ji

5) Skew Symmetric Matrix : A square matrix is called skew symmetric matrix if
i)for all i and j , a_ij = - a_ji
ii) All diagonal elements are zero.

6) Upper Triangular Matrix : A triangular matrix whose elements below the leading diagonal are zero .

7) Orthogonal Matrix : AA^T = I

8) Matrix A^θ : = (conjugate of A)^T

9) Unitary Matrix : A^θA = I

10) Hermitian Matrix : i) a_ij = conjugate of a_ji
ii) A = A^θ

11) Idempotent Matrix : A² = A

12) Periodic matrix : A^k+1 = A

13) Nilpotent Matrix : A^k = 0,k>0

14) Involuntary Matrix : A² = I

15) Adjoint of A Matrix : adj. A is the transpose of the matrix of co-factors of any matrix A.

16) Inverse Matrix : A^-1 = 1/|A| adj. A

** Rank of Matrix :

The rank of matrix is 'r' if

a) It has at least one non-zero minor of order 'r'.

b) every minor of 'A' of order higher than 'r' is zero.

i.e. Rank = No. of non-zero rows in upper triangular matrix.

Rank of augmented matrix A^ defining any given set of linear equations and coefficient matrix C decides type of solution:

1) Consistent Equations : Rank A^ = Rank C

i) Unique solution : Rank A^ = Rank C = n, n => No. of unknowns

ii) Infinite Solution : Rank A^ Rank C = m, m < n 2) Inconsistent Equations : Rank A^ != Rank C No Solution Possible.

Rank of Matrix and Matrix Dependency :

Vectors A1, A2, ..An are said to be dependent if :

1) all vectors are of same order

2) n scalars a1,a2, ..an (at least one non-zero) exist as

a1.A1 + a2.A2 + ...+an.An = 0

Otherwise they are linearly independent.


**Rank of matrix A is maximum no. of linearly independent row vectors of A or no. of linearly independent column vectors of A
Hence A and A^T has same rank..

** m vectors with n components each are linearly independent if if the matrix with these vectors as row vectors has rank m, but they are linearly dependent if that rank is less than m.

**TERMS YOU SHOULD ALSO KNOW : **

LINEAR SYSTEM, HOMOGENEOUS AND NON-HOMOGENEOUS SYSTEMS, STOCHASTIC MATRIX, TRIVIAL SOLUTION, GAUSS ELIMINATION, ROW EQUIVALENT SYSTEMS, ECHELON FORM, VECTOR SPACE, ROW SPACE, COLUMN SPACE, CRAMER'S RULE, MINORS AND CO-FACTORS, GAUSS-JORDAN ELIMINATION.