Linear Algebra
Exam Duration: 45 Mins Total Questions : 30
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
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
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
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
The product [P][Q]T of the following two matrices [P] and [Q] is
[P] = \(\begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix}\) and [Q] = \( \begin{bmatrix} 4 & 8 \\ 9 & 2 \end{bmatrix}\)
- (a)
\(\begin{bmatrix} 32 & 24 \\ 56 & 46 \end{bmatrix}\)
- (b)
\(\begin{bmatrix} 46 & 56 \\ 24 & 32 \end{bmatrix}\)
- (c)
\(\begin{bmatrix} 35 & 22 \\ 61 & 42 \end{bmatrix}\)
- (d)
\(\begin{bmatrix} 32 & 56 \\ 24 & 46 \end{bmatrix}\quad \)
The matrix [A] = \(\begin{bmatrix} 2 & 1 \\ 4 & -1 \end{bmatrix}\) is decomposed into a product of a lower triangular matrix L and an upper triangular matrux U.The properly decomposed L and U matrices respcetively are
- (a)
\(\begin{bmatrix} 1 & 0 \\ 4 & -1 \end{bmatrix}\) and \(\begin{bmatrix} 1 & 1 \\ 0 & -2 \end{bmatrix}\)
- (b)
\(\begin{bmatrix} 2 & 0 \\ 4 & -1 \end{bmatrix}\) and \(\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\)
- (c)
\(\begin{bmatrix} 1 & 0 \\ 4 & 1 \end{bmatrix}\) and \(\begin{bmatrix} 2 & 1 \\ 0 & -1 \end{bmatrix}\)
- (d)
\(\begin{bmatrix} 2 & 0 \\ 4 & -3 \end{bmatrix}\) and \(\begin{bmatrix} 1 & 0.5 \\ 0 & 1 \end{bmatrix}\)
| A | is a square matrix which is neither symmetric nor skew- symmetric and [A]T is its transpose. The sum and difference of these matrices and defined as [S] = [A]+[A]T and [D] = [A]-[A]T, respectively .Which of the following statements is true?
- (a)
Both [S] and [D] are symmetric
- (b)
Both [S] and [D] are skew-symmetric
- (c)
[S] is skew symmetirc and [D] is symmetric
- (d)
[S] is symmetric and [D] is skew-symmetric
A is square matrix and B is skew - symmetric , if
- (a)
BT = - B
- (b)
BT = B
- (c)
B-1 = B
- (d)
B-1 = BT
The product of matrices (PQ)-1P is
- (a)
p-1
- (b)
Q-1
- (c)
p-1Q-1P
- (d)
PQP-1
Consider the matrices X(4\(\times\)3), Y(4\(\times\)3) and P(4\(\times\)3).The order of [(P(XTY)-1PT)]T will be
- (a)
(2\(\times\)2)
- (b)
(3\(\times\)3)
- (c)
(4\(\times\)3)
- (d)
(3\(\times\)4)
Let A,B,C and D be n\(\times\)n matrices each with non-zero determinant. If ABCD = 1, then B-1 is
- (a)
D-1C-1A-1
- (b)
CDA
- (c)
ADC
- (d)
Does not necessarily exist
given A = \(\begin{bmatrix} 2 &\quad 0 \quad \quad 0\quad -1 \\ 0 & 1\quad \quad 0\quad 0 \\ \quad 0\quad & 0\quad \quad 3\quad 0 \\ -1 & 0\quad \quad 0\quad 4 \end{bmatrix}\). Sum of the eigen values of the matrix A is
- (a)
10
- (b)
-10
- (c)
24
- (d)
22
The eigen values of a skew - symmetric matrix are
- (a)
always zero
- (b)
always pure imaginary
- (c)
Either zero or pure imaginary
- (d)
always real
The eigen values of the following matrix are \(\begin{bmatrix} 1 & 3\quad 5 \\ -3 & -1\quad 6 \\ 0 & 0\quad 3 \end{bmatrix}\)
- (a)
3, 3+5j and 6-j
- (b)
-6+5j, 3+j and 3-j
- (c)
3+j,3-j and 5+j
- (d)
3, -1+3j and -1-3j
The eigen values and the correcponding eigen vectors of a(2\(\times\)2) matrix are given by
Eigen value | Eigen vector |
\(\lambda_1\) = 8 | V1 = \(\begin{bmatrix} 1 & \\ 1 & \end{bmatrix}\) |
\(\lambda_2\) = 4 | V2 =\(\begin{bmatrix} 1 & \\ -1 & \end{bmatrix}\) |
The matrix is
- (a)
\(\begin{bmatrix} 6 & \quad 2\quad \\ 2 & \quad 6\quad \end{bmatrix}\)
- (b)
\(\begin{bmatrix} 4 & \quad 6\quad \\ 6 & \quad 4\quad \end{bmatrix}\)
- (c)
\(\begin{bmatrix} 2 & \quad 4\quad \\ 4 & \quad 2\quad \end{bmatrix}\)
- (d)
\(\begin{bmatrix} 4 & \quad 8\quad \\ 8 & \quad 4\quad \end{bmatrix}\)
For which value of x will be matrix given below become singular ?
\( \begin{bmatrix} 8 & x\quad 0 \\ 4 & 0\quad 2 \\ 12 & 6\quad 0 \end{bmatrix}\)
- (a)
4
- (b)
6
- (c)
8
- (d)
12
For a given matrix A = \(\left[ \begin{matrix} 4 & -2 \\ -2 & 1 \end{matrix} \right] \) eigenvalues are
- (a)
1 and 4
- (b)
-1 and 2
- (c)
0 and 5
- (d)
Can't be determined
Consider the following matrix A = \(\begin{bmatrix} 2 & 3 \\ x & y \end{bmatrix}\) . if the eigenvalues of A are 4 and 8, then
- (a)
x = 4 and y = 10
- (b)
x = 5 and y = 8
- (c)
x = -3 and y = 9
- (d)
x = -4 and y = 10
If a square matrix A is real and symmetric, then the eigenvalues
- (a)
are always real
- (b)
are always real and positive
- (c)
are always real and non-negative
- (d)
occur in complex conjucate pairs
Let a = \(\begin{bmatrix} 2 & -0.1 \\ 0 & 3 \end{bmatrix}\quad \) and A-1 = \(\begin{bmatrix} \frac { 1 }{ 2 } & a \\ 0 & b \end{bmatrix}\) . Then , ( a + b ) is equal to
- (a)
\(\frac{7}{20}\)
- (b)
\(\frac{3}{20}\)
- (c)
\(\frac{19}{60}\)
- (d)
\(\frac{11}{20}\)
For the matrix \(\begin{bmatrix} 4 & 1 \\ 1 & 4 \end{bmatrix}\) the eigen values are
- (a)
3 and -3
- (b)
-3 and -5
- (c)
3 and 5
- (d)
5 and 0
For the set of equations
x1 + 2x2+x3+4x4 = 2
3x1+6x2+3x3+12x4 =6
Which of the following statements is true ?
- (a)
Only the trivial solution x1=x2 = x3= x4= 0 exists
- (b)
There are no solution
- (c)
A unique non - trivial solution exists
- (d)
Multiple non - trivial solution exists
Consider the following system of equations x1, x2 and x3
2x1-x2+3x3 = 1
3x1-2x2+5x3 = 2
-x1-4x2+x3 = 3
This system of equations has
- (a)
no solution
- (b)
an unique solution
- (c)
more than one but a finite number of solutions
- (d)
an infinite number of solutions
Given, the matrix \(\begin{bmatrix} -4 & 2 \\ 4 & 3 \end{bmatrix}\) , the eigen vector is
- (a)
\(\left[ \begin{matrix} 3 \\ 2 \end{matrix} \right] \)
- (b)
\(\left[ \begin{matrix} 4 \\ 3 \end{matrix} \right] \)
- (c)
\(\left[ \begin{matrix} 2 \\ -1 \end{matrix} \right] \)
- (d)
\(\left[ \begin{matrix} -1 \\ 2 \end{matrix} \right] \)
Consider the matrix as given below :
\(\begin{bmatrix} 1 & 2\quad 3 \\ 0 & 4\quad 7 \\ 0 & 0\quad 3 \end{bmatrix}\) . Which one of the following provides the correct values of eigen values of the matrix?
- (a)
1,4 and 3
- (b)
3,7 and 3
- (c)
7, 3 and 2
- (d)
1, 2 and 3
Eigen values of a matrix S = \(\begin{bmatrix} 3 & 2 \\ 2 & 3 \end{bmatrix}\) are 5 and 1.Wht are the eigen values of the m,atrix S2 = SS?
- (a)
1 and 25
- (b)
6 and 4
- (c)
5 and 1
- (d)
2 and 10
The three characteristic roots of the following matrix A = \( \begin{bmatrix} 1 & 2\quad 3 \\ 0 & 2\quad 3 \\ 0 & 0\quad 2 \end{bmatrix}\) are
- (a)
1 , 2 and 3
- (b)
1, 2 and 2
- (c)
1, 0 and 0
- (d)
0, 2 and 3
Eigen values of the matrix \(\begin{bmatrix} -1 & 4 \\ 4 & -1 \end{bmatrix}\) are
- (a)
3 and-5
- (b)
-3 and 5
- (c)
-3 and -5
- (d)
3 and 5
If A = \(\begin{bmatrix} 1 & 3\quad \quad 5 \\ 0 & 2\quad -1 \\ 0 & 0\quad \quad 3 \end{bmatrix}\) , then eigen values of the matrix I +A+A2, where I denote the identity by matrix, are
- (a)
3, 7 and 11
- (b)
3,7 and 12
- (c)
3,7 and 13
- (d)
3, 9 and 16
The following simultaneous equations
x+y+z=3
x+2y+3z=4
x+4y+kz=6
will not have a unique solution for k is equal to
- (a)
0
- (b)
5
- (c)
6
- (d)
7