Engineering Mathematics - Linear Algebra
Exam Duration: 45 Mins Total Questions : 30
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
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
The product of matrices (PQ)-1P is
- (a)
p-1
- (b)
Q-1
- (c)
p-1Q-1P
- (d)
PQP-1
The inverse of the matrix \(\begin{bmatrix} 3+2i & i \\ -i & 3-2i \end{bmatrix}\) is
- (a)
\(\frac{1}{12}\) \(\begin{bmatrix} 3+2i & -i \\ i & 3-2i \end{bmatrix}\)
- (b)
\(\frac{1}{12}\) \(\begin{bmatrix} 3-2i & -i \\ i & 3+2i \end{bmatrix}\)
- (c)
\(\frac{1}{14}\) \(\begin{bmatrix} 3+2i & -i \\ i & 3-2i \end{bmatrix}\)
- (d)
\(\frac{1}{14}\) \(\begin{bmatrix} 3-2i & -i \\ i & 3+2i \end{bmatrix}\)
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
How many equations does the following system of linear equations have ?
-x+5y = -1
x-y = 2
x+3y = 3
- (a)
Infinitely many
- (b)
Two distinct solutions
- (c)
Unique
- (d)
None of these
For the matrix P = \(\begin{bmatrix} 3 & -2\quad 2 \\ 0 & -2\quad 1 \\ 0 & 0\quad 1 \end{bmatrix}\), one of the eigen values is equal to -2. Which of the following is eigen vector?
- (a)
\(\quad \begin{bmatrix} 3 & \\ -2 & \\ 1 & \end{bmatrix}\)
- (b)
\(\quad \begin{bmatrix} -3 & \\ 2 & \\ -1 & \end{bmatrix}\)
- (c)
\(\quad \begin{bmatrix} 1 & \\ -2 & \\ 3 & \end{bmatrix}\)
- (d)
\(\quad \begin{bmatrix} 2 & \\ 5 & \\ 0 & \end{bmatrix}\)
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)
\(\begin{bmatrix} 4 & \quad 8\quad \\ 8 & \quad 4\quad \end{bmatrix}\)
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
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)
\(\frac{11}{20}\)
The sum of the eigenvalues of the matrix given below is \(\left[ \begin{matrix} 1 & 1 & 3 \\ 1 & 5 & 1 \\ 3 & 1 & 1 \end{matrix} \right] \)
- (a)
5
- (b)
7
- (c)
9
- (d)
18
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
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)
\(\begin{bmatrix} \frac { 1 }{ 4 } & 0\quad \quad 0\quad \quad 0 \\ 0 & \frac { 1 }{ 4 } \quad \quad 0\quad \quad 0 \\ 0 & 0\quad \quad \frac { 1 }{ 4 } \quad 0 \\ 0 & 0\quad \quad 0\quad \quad \frac { 1 }{ 4 } \end{bmatrix}\)
Rank of the matrix \(\begin{bmatrix} 1 & 4\quad \quad 8\quad \quad 7 \\ 0 & 0\quad \quad 3\quad \quad 0 \\ 4 & 3\quad \quad 2\quad \quad 1 \\ 3\quad & 12\quad \quad 24\quad 21 \end{bmatrix}\) is
- (a)
3
- (b)
1
- (c)
2
- (d)
4
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
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
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
For what value of a if any , will be following system of equations in x, y and z have a solution?
2x+3y = 4
x+y+z = 4
x+2y-z = a
- (a)
any real number
- (b)
0
- (c)
1
- (d)
There is no such value
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)
\(\left[ \begin{matrix} -1 \\ 2 \end{matrix} \right] \)
All the four entries of the (2\(\times\)2) P = \(\begin{bmatrix} { P }_{ 11 } & { P }_{ 12 } \\ { P }_{ 21 } & { P }_{ 22 } \end{bmatrix}\) are non - zero and one of its eigen values is zero. Which of the following statements is true?
- (a)
P11P22-P12P21 = -1
- (b)
P11P22-P12P21 = -1
- (c)
P11P22-P12P21 = 0
- (d)
P11P22+P12P21 = 0
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 eigen values of the matrix \(\begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}\) are written in the form \(\left[ \begin{matrix} 1 \\ a \end{matrix} \right] \)and \(\left[ \begin{matrix} 1 \\ b \end{matrix} \right] \), what (a+b)?
- (a)
0
- (b)
\(\frac{1}{2}\)
- (c)
2
- (d)
infinite
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
For what value of a and b, the following simultaneous equations have an infinite number of solutions?
x+y+z=5, x+3y+3z=9 and x+2y+az=b
- (a)
2 and 7
- (b)
3 and 8
- (c)
8 and 3
- (d)
7 and 2