Linear Algebra
Exam Duration: 45 Mins Total Questions : 30
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
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)
\(\begin{bmatrix} 1+2n & -4n \\ 2 & 1-2n \end{bmatrix}\)
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
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)
\(\begin{bmatrix} \frac{1}{5} & 0\quad -\frac{1}{2} \\ 0 & \frac{1}{3}\quad 0 \\ -\frac{1}{2} & 0\quad 1 \end{bmatrix}\)
Eigen roots of the matrix \(\begin{bmatrix} 1 & 0\quad \quad 0\quad \quad 0 \\ 100 & 1\quad \quad 0\quad \quad 0 \\ 100 & 200\quad 1\quad \quad 0 \\ 100 & 200\quad 300\quad 1 \end{bmatrix}\) is
- (a)
100
- (b)
200
- (c)
1
- (d)
300
| 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
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)
\(\frac{1}{14}\) \(\begin{bmatrix} 3-2i & -i \\ i & 3+2i \end{bmatrix}\)
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
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
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
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 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 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
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
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
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
The system of equations
x+y+z = 6
x+4y+6z = 20
x+4y+\(\lambda\)z = \(\mu\)
has no solution for values of \(\lambda\) and \(\mu\) given by
- (a)
\(\lambda\) = 6 and \(\mu\) = 20
- (b)
\(\lambda\) = 6 and \(\mu\) \(\neq\) 20
- (c)
\(\lambda\) \(\neq\) 6 and \(\mu\) = 20
- (d)
\(\lambda\) \(\neq\) 6 and \(\mu\) \(\neq\) 20
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
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
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
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)
\(\left[ \begin{matrix} 1 \\ -1 \end{matrix} \right] \)
th eeigen values of th ematrix [ P ] = \(\begin{bmatrix} 4 & 5 \\ 2 & -5 \end{bmatrix}\) are
- (a)
-7 and 8
- (b)
-6 and 5
- (c)
3 and 4
- (d)
1 and 2
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
Eigen values of the matrix \(\begin{bmatrix} 0 & 0\quad \alpha \\ 0 & 0\quad 0 \\ 0 & 0\quad 0 \end{bmatrix}\) , \(\alpha\) \(\neq\) 0 are
- (a)
(0,0,\(\alpha\))
- (b)
(\(\alpha\),0,0)
- (c)
(0,0,1)
- (d)
(0,\(\alpha\),0)
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
An eigen vector of \(\begin{bmatrix}1&1&0\\0&2&2\\0&0&3 \end{bmatrix}\) is
- (a)
[-1 1]T
- (b)
[1 2 1]T
- (c)
[1 -1 2]T
- (d)
[2 1 -1]T