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Mathematics MCQ - Vectors and Matrices

Linear Algebra GATE Syllabus

Matrices, determinants, system of linear equations, eigenvalues and eigenvectors, LU decomposition.

1 Rank of the matrix A =

0	0	0	0
4	2	3	0
1	0	0	0
4	0	3	0

Determinant A = 0 ⇒

4	2	3
1	0	0
4	0	3

Hence rank of A=3

2 A set of linear equations is represented by the matrix equation Ax = b. The necessary condition for the existence of a solution for this system is

x=A^-1b

3 The system of linear equations
(4d - 1)x +y + z = 0
- y + z = 0
(4d - 1) z = 0
has a non-trivial solution, if d equals

The system of homogeneous linear equations has a non-trivial solution if

4d-1	1	1
0	-1	1
0	0	4d-1

=

0

=> -(4d-1)² = 0 => d = 1/4

4 The rank of a 3 x 3 matrix C (= AB), found by multiplying a non-zero column matrix A of size 3 x 1 and a non-zero row matrix B of size 1 x 3, is

Let

A

=

a1

b1

c1

;

B

=

a2

b2

c2

;

∴

C

=

a1a2		a1b2		a1c2
b1a2		b1b2		b1c2
c1a1		c1b2		c1c2

5 If A and B are square matrices of size n x n, then which of the following statement is not true?

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Day 3 / 10

Question Spotlight

Question:

Rank of the matrix A = 0 0 0 0 4 2 3 0 1 0 0 0 4 0 3 0

Answer:

Determinant A = 0 ⇒ 4 2 3 1 0 0 4 0 3 Hence rank of A=3

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