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
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