### Engineering Mathematics - Linear Algebra

#### Question - 1

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

• A A must be invertible
• B b must be  linearly dependent on the columns of A
• C b must be linearly independent of the columns of A
• D None of the above

#### Question - 2

In the matrix equation px = q, which of the following is a neccessary condition for the existence of atleast one  solution for the unknown vector x ?

• A  Augmented [pq] must have the same rank a matrix p
• B Vector q must have only non-zero elementy
• C Matrix p must be singular
• D Matrix p must be square

#### Question - 3

Consider a non-homogeneous system of linear equations representing mathematically an over determined system. Such a system will be

• A consistent, having an unique solution
• B consistent , having many solutions
• C inconsistent , having an unique solution
• D inconsistent, having no solution

#### Question - 4

nulity of the matrix A = $\begin{bmatrix} -1 & 4\quad 2 \\ 1 & 3\quad 2 \\ -2 & 1\quad 0 \\ 2 & 6\quad 4 \end{bmatrix}$  is

• A 1
• B 2
• C 3
• D  4

#### Question - 5

If A and B are real  symmetric matrices of size n $\times$ n , then,

• A AAT = I
• B A = A-1
• C AB = BA
• D  (AB)T = BTAT

#### Question - 6

Consider th efollowing statements:
S1: Sum of the two singular matrices may be non - singular
S2: Sum of the  two non - singular n $\times$ n matrices may be singular.
Which of the following statements is correct?

• A S1 and S2 both are correct
• B S1 is true and S2 is false
• C S1 is false and S2 is true
• D S1 and S2 both are false

#### Question - 7

Let A = (aij) be an n - rowed square matrix and I12 be the matrix obtained by interchanging th efirst and second rows of the n-owned identify matrix.Then , AI12 is such that its first

• A row is the same as its second row
• B row is the same as the second row of A
• C column is the same as the second row of A
• D row is all zeros

#### Question - 8

If A = $\begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix}$  , then An is equal to

• A $\begin{bmatrix} 3n & -4n \\ n & -n \end{bmatrix}$
• B $\begin{bmatrix} 2+n & 5-n \\ n & -n \end{bmatrix}$
• C $\begin{bmatrix} 3^ n & (-4)^ n \\ 1^ n & (-1)^ n \end{bmatrix}$
• D $\begin{bmatrix} 1+2n & -4n \\ 2 & 1-2n \end{bmatrix}$

#### Question - 9

If A,B and C are square matrices of the same order, then (ABC)-1 is equal to

• A C-1A-1B-1
• B C-1B-1A-1
• C A-1B-1C-1
• D A-1C-1B-1

#### Question - 10

Determinant of the matrix  $\begin{bmatrix} 5 & 3\quad 2 \\ 1 & 2\quad 6 \\ 3 & 5\quad 10 \end{bmatrix}$  is

• A -76
• B   -28
• C   28
• D 72