Engineering Mathematics - 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
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
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)
\(\begin{bmatrix} 1+2n & -4n \\ 2 & 1-2n \end{bmatrix}\)
If A = \(\begin{bmatrix} 5 & 0\quad 2 \\ 0 & 3\quad 0 \\ 2 & 0\quad 1 \end{bmatrix}\) . Then, inverse of A is
- (a)
\(\begin{bmatrix} 1 & 0\quad -2 \\ 0 & \frac{1}{3}\quad 0 \\ -2 & 0\quad 5 \end{bmatrix}\)
- (b)
\(\begin{bmatrix} 5 & 0\quad 2 \\ 0 & -\frac{1}{3}\quad 0 \\ 2 & 0\quad 1 \end{bmatrix}\)
- (c)
\(\begin{bmatrix} \frac{1}{5} & 0\quad \frac{1}{2} \\ 0 & \frac{1}{3}\quad 0 \\ \frac{1}{2} & 0\quad 1 \end{bmatrix}\)
- (d)
\(\begin{bmatrix} \frac{1}{5} & 0\quad -\frac{1}{2} \\ 0 & \frac{1}{3}\quad 0 \\ -\frac{1}{2} & 0\quad 1 \end{bmatrix}\)
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
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
Let AX = b be a system of linear equations, where A is an n\(\times\)n matrix, b is m\(\times\)1 column vector and X is a n\(\times\)1 column vector of unknowns. Which of the following statements is false?
- (a)
The system has a solution if and only idf both A and the augmented matrix [A b] have the same rank
- (b)
If m<n and b is the zero vector, then the system has infinitely many solutions
- (c)
If m = n and b is a non- zero vector, then the system has a unique solution
- (d)
The system will have only a trivial solution when m= n,b is zero vector and rank(A) = n
The following system of equations
3x+2y+z = 4
x-2y+z = 2
-2x+2z = 5
- (a)
no solution
- (b)
an unique solution
- (c)
multiple solution
- (d)
an inconsistency
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 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)
\(\begin{bmatrix} 4 & \quad 8\quad \\ 8 & \quad 4\quad \end{bmatrix}\)
Multiplication of matrices E and F is G. matrices E and G are
E = \(\begin{bmatrix} cos\theta & -sin\theta \quad 0 \\ sin\theta & cos\theta \quad \quad 0 \\ 0 & 0\quad \quad \quad 1 \end{bmatrix}\) and G = \(\begin{bmatrix} 1 & 0\quad 0 \\ 0 & 1\quad 0 \\ 0 & 0\quad 1 \end{bmatrix}\)
What is the matrix F ?
- (a)
\(\begin{bmatrix} cos\theta & -sin\theta \quad 0 \\ sin\theta & cos\theta \quad \quad 0 \\ 0 & 0\quad \quad \quad 1 \end{bmatrix}\)
- (b)
\(\begin{bmatrix} sin\theta & cos\theta \quad 0 \\ -cos\theta & sin\theta \quad \quad 0 \\ 0 & 0\quad \quad \quad 1 \end{bmatrix}\)
- (c)
\(\begin{bmatrix} cos\theta & sin\theta \quad 0 \\ -sin\theta & cos\theta \quad \quad 0 \\ 0 & 0\quad \quad \quad 1 \end{bmatrix}\)
- (d)
\(\begin{bmatrix} sin\theta & -cos\theta \quad 0 \\ cos\theta & sin\theta \quad \quad 0 \\ 0 & 0\quad \quad \quad 1 \end{bmatrix}\)
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
The rank of a 3 x 3 matrix (C=AB0, 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
- (a)
0
- (b)
1
- (c)
2
- (d)
3
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
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
Consider the matrix P = \(\begin{bmatrix} 0 & 1 \\ -2 & -3 \end{bmatrix}\) . The value of eP is
- (a)
\(\begin{bmatrix} 2{ e }^{ -2 }-3{ e }^{ -1 } & { e }^{ -1 }-{ e }^{ -2 } \\ 2{ e }^{ -2 }-2{ e }^{ -1 } & { 5{ e }^{ -2 }-{ e }^{ -1 } } \end{bmatrix}\)
- (b)
\(\begin{bmatrix} { e }^{ -1 }{ +e }^{ -2 } & 2{ e }^{ -2 }-{ e }^{ -1 } \\ 2{ e }^{ -1 }-4{ e }^{ 2 } & 3{ e }^{ -1 }+2{ e }^{ -2 } \end{bmatrix}\)
- (c)
\(\begin{bmatrix} 5{ e }^{ -2 }{ -e }^{ -1 } & 3{ e }^{ -1 }-{ e }^{ -2 } \\ 2{ e }^{ -2 }-6{ e }^{- 1 } & 4{ e }^{ -2 }+{ e }^{ -1 } \end{bmatrix}\)
- (d)
\(\begin{bmatrix} 2{ e }^{ -1 }{ -e }^{ -2 } & { e }^{ -1 }-{ e }^{ -2 } \\ 2{ e }^{ -1 }+2{ e }^{ -2 } & { -e }^{ -1 }+2{ e }^{ -2 } \end{bmatrix}\)
Given an orthogonal matrix A = \(\begin{bmatrix} 1 & 1\quad \quad 1\quad \quad 1 \\ 1 & -1\quad \quad 0\quad \quad 0 \\ 1 & -1\quad \quad 0\quad \quad 0 \\ 0 & 0\quad \quad 1\quad -1 \end{bmatrix}\)
[AAT]-1 is
- (a)
\(\begin{bmatrix} \frac { 1 }{ 4 } & 0\quad \quad 0\quad \quad 0 \\ 0 & \frac { 1 }{ 4 } \quad \quad 0\quad \quad 0 \\ 0 & 0\quad \quad \frac { 1 }{ 2 } \quad 0 \\ 0 & 0\quad \quad 0\quad \quad \frac { 1 }{ 2 } \end{bmatrix}\)
- (b)
\(\begin{bmatrix} \frac { 1 }{ 2 } & 0\quad \quad 0\quad \quad 0 \\ 0 & \frac { 1 }{2 } \quad \quad 0\quad \quad 0 \\ 0 & 0\quad \quad \frac { 1 }{ 2 } \quad 0 \\ 0 & 0\quad \quad 0\quad \quad \frac { 1 }{ 2 } \end{bmatrix}\)
- (c)
\(\quad \begin{bmatrix} 1 & 0\quad 0\quad 0 \\ 0 & 1\quad 0\quad 0 \\ 0 & 0\quad 1\quad 0 \\ 0 & 0\quad 0\quad 1 \end{bmatrix}\)
- (d)
\(\begin{bmatrix} \frac { 1 }{ 4 } & 0\quad \quad 0\quad \quad 0 \\ 0 & \frac { 1 }{ 4 } \quad \quad 0\quad \quad 0 \\ 0 & 0\quad \quad \frac { 1 }{ 4 } \quad 0 \\ 0 & 0\quad \quad 0\quad \quad \frac { 1 }{ 4 } \end{bmatrix}\)
A is m\(\times\)n full matrix with m>n and I is an identity matrix .Let matrix AT = (ATA)-1AT. Then , which one of the following statements is false?
- (a)
ATA =A-1
- (b)
(AAT)2 = I
- (c)
ATA =I
- (d)
AATA =A
A is a 3\(\times\)4 real matrix and Ax =b is inconsistent system of equations.The highest possible rank of A is
- (a)
1
- (b)
2
- (c)
3
- (d)
4
How many of the following matrices have an eigen value 1 ?
\(\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\) , \(\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}\) , \(\begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix}\) and \(\begin{bmatrix} -1 & 0 \\ 1 & -1 \end{bmatrix}\)
- (a)
One
- (b)
Two
- (c)
Three
- (d)
Four
Which of the following is an eigen vector of the matrix \(\begin{bmatrix} 5 & 0\quad 0\quad 0 \\ 0 & 5\quad 0\quad 0 \\ 0 & 0\quad 2\quad 1 \\ 0 & 0\quad 3\quad 1 \end{bmatrix}\) ?
- (a)
\(\left[ \begin{matrix} 1 \\ \begin{matrix} -2 \\ \begin{matrix} 0 \\ 0 \end{matrix} \end{matrix} \end{matrix} \right] \)
- (b)
\(\left[ \begin{matrix} 0 \\ \begin{matrix} 0 \\ \begin{matrix} 1 \\ 0 \end{matrix} \end{matrix} \end{matrix} \right] \)
- (c)
\(\left[ \begin{matrix} 1 \\ \begin{matrix} 0 \\ \begin{matrix} 0 \\ -2 \end{matrix} \end{matrix} \end{matrix} \right] \)
- (d)
\(\left[ \begin{matrix} 1 \\ \begin{matrix} -1 \\ \begin{matrix} 2 \\ 1 \end{matrix} \end{matrix} \end{matrix} \right] \)
The number of linearly independent eigen vectors of \(\begin{bmatrix} 2 & 1 \\ 0 & 2 \end{bmatrix}\) is
- (a)
0
- (b)
1
- (c)
2
- (d)
infinite
One of the eigen vectors of the matrix \(\begin{vmatrix} 2 & 2 \\ 1 & 3 \end{vmatrix}\) is
- (a)
\(\left[ \begin{matrix} 2 \\ -1 \end{matrix} \right] \)
- (b)
\(\left[ \begin{matrix} 2 \\ 1 \end{matrix} \right] \)
- (c)
\(\left[ \begin{matrix} 4 \\ 1 \end{matrix} \right] \)
- (d)
\(\left[ \begin{matrix} 1 \\ -1 \end{matrix} \right] \)
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