Engineering Mathematics - 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
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 = \(\begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix}\) , 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
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
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 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
The system of linear equations 4x+2y = 7 and 2x+y = 6 has
- (a)
an unique solution
- (b)
no solution
- (c)
an infinite number of solutions
- (d)
exactly two distinct solutions
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
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
The matrix \(\left[ \begin{matrix} 1 & 2 & 4 \\ 3 & 0 & 6 \\ 1 & 1 & P \end{matrix} \right] \) has one eigenvalue to 3. The sum of the two eigenvalues is
- (a)
P
- (b)
P - 1
- (c)
P - 2
- (d)
P - 3
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)
\(\frac{11}{20}\)
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)
\(\begin{bmatrix} \frac { 1 }{ 4 } & 0\quad \quad 0\quad \quad 0 \\ 0 & \frac { 1 }{ 4 } \quad \quad 0\quad \quad 0 \\ 0 & 0\quad \quad \frac { 1 }{ 4 } \quad 0 \\ 0 & 0\quad \quad 0\quad \quad \frac { 1 }{ 4 } \end{bmatrix}\)
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
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
Consider the system of equations given below?
x+y = 2
2x+2y = 5
This system has
- (a)
One solution
- (b)
no solution
- (c)
infinite solution
- (d)
four solution
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
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
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
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
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)
\(\left[ \begin{matrix} 1 \\ \begin{matrix} -1 \\ \begin{matrix} 2 \\ 1 \end{matrix} \end{matrix} \end{matrix} \right] \)
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
The Characteristic equation of a (3\(\times\)3) matrix P is defined as a ( \(\lambda\) ) = | \(\lambda\) I - P | = \(\lambda\)3+\(\lambda\) 2+2\(\lambda\) + I = 0.If / denotes identity matrix, then the inverse of P will be
- (a)
P2+P+2I
- (b)
P2+P+I
- (c)
-(P2+P+I)
- (d)
-( P2+P+2I)
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
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
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