Engineering Mathematics - Linear Algebra
Exam Duration: 45 Mins Total Questions : 30
In the matrix equation px = q, which of the following is a neccessary condition for the existence of atleast one solution for the unknown vector x ?
- (a)
Augmented [pq] must have the same rank a matrix p
- (b)
Vector q must have only non-zero elementy
- (c)
Matrix p must be singular
- (d)
Matrix p must be square
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
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)
\(\begin{bmatrix} 1+2n & -4n \\ 2 & 1-2n \end{bmatrix}\)
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)
\(\begin{bmatrix} 32 & 56 \\ 24 & 46 \end{bmatrix}\quad \)
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
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 \(\begin{bmatrix} 1 & 0.5 \\ 0 & 1 \end{bmatrix}\)
| 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 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)
\(\frac{1}{3}\) \(\begin{bmatrix} -7 & -2 \\ -5 & -1 \end{bmatrix}\)
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
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
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
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)
\(\begin{bmatrix} 4 & \quad 8\quad \\ 8 & \quad 4\quad \end{bmatrix}\)
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 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
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
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)
\(\frac{11}{20}\)
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
Consider the matrix P = \(\begin{bmatrix} 0 & 1 \\ -2 & -3 \end{bmatrix}\) . The value of eP is
- (a)
\(\begin{bmatrix} 2{ e }^{ -2 }-3{ e }^{ -1 } & { e }^{ -1 }-{ e }^{ -2 } \\ 2{ e }^{ -2 }-2{ e }^{ -1 } & { 5{ e }^{ -2 }-{ e }^{ -1 } } \end{bmatrix}\)
- (b)
\(\begin{bmatrix} { e }^{ -1 }{ +e }^{ -2 } & 2{ e }^{ -2 }-{ e }^{ -1 } \\ 2{ e }^{ -1 }-4{ e }^{ 2 } & 3{ e }^{ -1 }+2{ e }^{ -2 } \end{bmatrix}\)
- (c)
\(\begin{bmatrix} 5{ e }^{ -2 }{ -e }^{ -1 } & 3{ e }^{ -1 }-{ e }^{ -2 } \\ 2{ e }^{ -2 }-6{ e }^{- 1 } & 4{ e }^{ -2 }+{ e }^{ -1 } \end{bmatrix}\)
- (d)
\(\begin{bmatrix} 2{ e }^{ -1 }{ -e }^{ -2 } & { e }^{ -1 }-{ e }^{ -2 } \\ 2{ e }^{ -1 }+2{ e }^{ -2 } & { -e }^{ -1 }+2{ e }^{ -2 } \end{bmatrix}\)
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
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
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)
\(\left[ \begin{matrix} 1 \\ \begin{matrix} -1 \\ \begin{matrix} 2 \\ 1 \end{matrix} \end{matrix} \end{matrix} \right] \)
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
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)
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
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