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
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
Let A = (aij) be an n - rowed square matrix and I12 be the matrix obtained by interchanging th efirst and second rows of the n-owned identify matrix.Then , AI12 is such that its first
- (a)
row is the same as its second row
- (b)
row is the same as the second row of A
- (c)
column is the same as the second row of A
- (d)
row is all zeros
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
The matrix [A] = \(\begin{bmatrix} 2 & 1 \\ 4 & -1 \end{bmatrix}\) is decomposed into a product of a lower triangular matrix L and an upper triangular matrux U.The properly decomposed L and U matrices respcetively are
- (a)
\(\begin{bmatrix} 1 & 0 \\ 4 & -1 \end{bmatrix}\) and \(\begin{bmatrix} 1 & 1 \\ 0 & -2 \end{bmatrix}\)
- (b)
\(\begin{bmatrix} 2 & 0 \\ 4 & -1 \end{bmatrix}\) and \(\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\)
- (c)
\(\begin{bmatrix} 1 & 0 \\ 4 & 1 \end{bmatrix}\) and \(\begin{bmatrix} 2 & 1 \\ 0 & -1 \end{bmatrix}\)
- (d)
\(\begin{bmatrix} 2 & 0 \\ 4 & -3 \end{bmatrix}\) and \(\begin{bmatrix} 1 & 0.5 \\ 0 & 1 \end{bmatrix}\)
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]
The product of matrices (PQ)-1P is
- (a)
p-1
- (b)
Q-1
- (c)
p-1Q-1P
- (d)
PQP-1
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)
\(\frac{1}{3}\) \(\begin{bmatrix} -7 & -2 \\ -5 & -1 \end{bmatrix}\)
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
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
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)
\(\quad \begin{bmatrix} 2 & \\ 5 & \\ 0 & \end{bmatrix}\)
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)
\(\begin{bmatrix} sin\theta & -cos\theta \quad 0 \\ cos\theta & sin\theta \quad \quad 0 \\ 0 & 0\quad \quad \quad 1 \end{bmatrix}\)
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
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 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
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
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
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
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 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
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
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