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 = \(\begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix}\) , 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,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
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
If R = \(\begin{bmatrix} 1 & \quad 0\quad -1 \\ 2 & \quad 1\quad -1 \\ 2 & \quad 3\quad \quad 2 \end{bmatrix}\), then the top rpw of R-1 is
- (a)
[5 6 4]
- (b)
[5 -3 1]
- (c)
[2 0 -1]
- (d)
[2 -1 1/2]
| 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
The inverse of the 2\(\times\)2 matrix \(\begin{bmatrix} 1 & 2 \\ 5 & 7 \end{bmatrix}\) is
- (a)
\(\frac{1}{3}\) \(\begin{bmatrix} -7 & 2 \\ 5 & -1 \end{bmatrix}\)
- (b)
\(\frac{1}{3}\) \(\begin{bmatrix} 7 & 2 \\ 5 & 1 \end{bmatrix}\)
- (c)
\(\frac{1}{3}\) \(\begin{bmatrix} 7 & -2 \\ -5 & 1 \end{bmatrix}\)
- (d)
\(\frac{1}{3}\) \(\begin{bmatrix} -7 & -2 \\ -5 & -1 \end{bmatrix}\)
The number of different n\(\times\)n symmetric matrices with each element being either zero or 1 is
( if power (2x) is same as 2x.)
- (a)
power(2,n)
- (b)
power(2,n2)
- (c)
power \([2, \frac{(n^2+n)}{2}]\)
- (d)
power \([2, \frac{(n^2-n)}{2}]\)
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
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)
\(\begin{bmatrix} 4 & \quad 8\quad \\ 8 & \quad 4\quad \end{bmatrix}\)
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)
\(\frac{4}{5}\)
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
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
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
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)
\(\left[ \begin{matrix} -1 \\ 2 \end{matrix} \right] \)
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 number of linearly independent eigen vectors of \(\begin{bmatrix} 2 & 1 \\ 0 & 2 \end{bmatrix}\) is
- (a)
0
- (b)
1
- (c)
2
- (d)
infinite
The three characteristic roots of the following matrix A = \( \begin{bmatrix} 1 & 2\quad 3 \\ 0 & 2\quad 3 \\ 0 & 0\quad 2 \end{bmatrix}\) are
- (a)
1 , 2 and 3
- (b)
1, 2 and 2
- (c)
1, 0 and 0
- (d)
0, 2 and 3
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
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)
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