Engineering Mathematics - Linear Algebra
Exam Duration: 45 Mins Total Questions : 30
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
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)
\(\begin{bmatrix} 32 & 56 \\ 24 & 46 \end{bmatrix}\quad \)
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)
\(\begin{bmatrix} \frac{1}{5} & 0\quad -\frac{1}{2} \\ 0 & \frac{1}{3}\quad 0 \\ -\frac{1}{2} & 0\quad 1 \end{bmatrix}\)
| 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
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
If the rank of a (5\(\times\)6) matrix Q is 4, then which one of the following statements is correct?
- (a)
Q will have four linearly independent rows and four linearly independent columns
- (b)
Q will have four linearly independent rows and five linearly independent columns
- (c)
QQT will be invertible
- (d)
QTQ will be invertible
Cnsider the system of simultaneous equations.
x+2y+z = 6
2x+y_2z = 6
x+y+z = 5
The system has
- (a)
unique solution
- (b)
infinite number of solutions
- (c)
no solution
- (d)
exactly two solution
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
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
For the matrix [M] = \(\begin{bmatrix} \frac { 3 }{ 5 } & \frac { 4 }{ 5 } \\ x & \frac { 3 }{ 5 } \end{bmatrix}\) , the transpose of the matrix is equal to the inverse of the matrix [M]T = [M]-1.the value of x is given by
- (a)
-\(\frac{4}{5} \)
- (b)
-\(\frac{3}{5}\)
- (c)
\(\frac{3}{5}\)
- (d)
\(\frac{4}{5}\)
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 a given matrix A =\(\left[ \begin{matrix} 2 & -2 & 3 \\ -2 & -1 & 6 \\ 1 & 2 & 0 \end{matrix} \right] \) one of the eigenvalues is 3. The other two eigenvalues are
- (a)
2 and -5
- (b)
3 and -5
- (c)
2 and 5
- (d)
3 and 5
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
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
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
Given matrix [ A ] = \(\begin{bmatrix} 4 & 2\quad 1\quad 3 \\ 6 & 3\quad 4\quad 7 \\ 2 & 1\quad 0\quad 1 \end{bmatrix}\) , the rank of the matrix is
- (a)
4
- (b)
3
- (c)
2
- (d)
1
The following system of equations
x1+x2+2x3 = 4
x1+2x2+3x3 = 2
X1+4x2+ax3 = 4
has an unique solution. The only possible value(s) of a is/are
- (a)
0
- (b)
Either 0 or 1
- (c)
0,1 or -1
- (d)
any real number other than 5
For the matrix \(\begin{bmatrix} 4 & 2 \\ 2 & 4 \end{bmatrix}\) , the eigen value corresponding to the eigen vector \(\begin{bmatrix}1 & 0 & 1 \\1 & 0 & 1 \end{bmatrix}\) is
- (a)
2
- (b)
4
- (c)
6
- (d)
8
What are the eigen values of the following (2\(\times\)2) matrix?
\(\begin{bmatrix} 2 & -1 \\ -4 & 5 \end{bmatrix}\)
- (a)
-1 and 1
- (b)
1 and 6
- (c)
2 and 5
- (d)
4 and -1
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
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 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
The trace and determinant of a (2\(\times\)2) matrix are known to be -2 and -35 , respectively .Its eigen values are
- (a)
-30 and -5
- (b)
-37 and -1
- (c)
-7 and 5
- (d)
17.5 and -2
The vector \(\left[ \begin{matrix} 1 \\ \begin{matrix} 2 \\ -1 \end{matrix} \end{matrix} \right] \) is an eigen vector of A = \(\begin{bmatrix} -2 & \quad 2\quad -3 \\ 2 & \quad 1\quad -6 \\ -1 & -2\quad \quad 0 \end{bmatrix}\) one of the eigen vector of A is
- (a)
1
- (b)
2
- (c)
5
- (d)
-1
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