Algebra of real matrices - Linear Algebra - Engineering Sciences Previous Year Questions

1

2007 · Engineering Sciences · Linear Algebra · Algebra of real matrices

Engineering Sciences (XE) 2007

Let \( M = \begin{pmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{pmatrix} \). Then the maximum number of linearly independent eigenvectors of \( M \) is

A

0

B

1

C

2

D

3

Open complete paper

2

2007 · Engineering Sciences · Linear Algebra · Algebra of real matrices

Engineering Sciences (XE) 2007

Let \( Ax = b \) be a system of \( m \) linear equations in \( n \) unknowns with \( m < n \) and \( b \neq 0 \). Then the system has

A

\(n - m \) solutions

B

either zero or infinitely many solutions

C

exactly one solution

D

\(n \) solutions

Open complete paper

3

2007 · Engineering Sciences · Linear Algebra · Algebra of real matrices

Engineering Sciences (XE) 2007

Let \( R \) be an \( n \times n \) nonsingular matrix. Let \( P \) and \( Q \) be two \( n \times n \) matrices such that \( Q = R^{-1} P R \). If \( x \) is an eigenvector of \( P \) corresponding to a nonzero eigenvalue \( \lambda \) of \( P \), then

A

\(Rx \) is an eigenvector of \(Q \) corresponding to the eigenvalue \(\lambda \) of \(Q \)

B

\(Rx \) is an eigenvector of \(Q \) corresponding to the eigenvalue \(\frac{1}{\lambda} \) of \(Q \)

C

\(R^{-1} x \) is an eigenvector of \(Q \) corresponding to the eigenvalue \(\frac{1}{\lambda} \) of \(Q \)

D

\(R^{-1} x \) is an eigenvector of \(Q \) corresponding to the eigenvalue \(\lambda \) of \(Q \)

Open complete paper

4

2007 · Engineering Sciences · Linear Algebra · Algebra of real matrices

Engineering Sciences (XE) 2007

The number of \( n \times n \) matrices that are simultaneously Hermitian, unitary and diagonal is

A

\(2^n \)

B

\(n^2 \)

C

\(2n \)

D

\(2 \)

Open complete paper

5

2007 · Engineering Sciences · Linear Algebra · Algebra of real matrices

Engineering Sciences (XE) 2007

Let \( M = \begin{pmatrix} 1 & b & a \\ 0 & 2 & c \\ 0 & 0 & 1 \end{pmatrix} \), where \( a, b, c \) are real numbers. Then \( M \) is diagonalizable

A

for all values of \(a, b, c \)

B

only when \(bc \neq a \)

C

only when \(b + c = a \)

D

only when \(bc = a \)

Open complete paper

6

2007 · Engineering Sciences · Linear Algebra · Algebra of real matrices

Engineering Sciences (XE) 2007

If the diagonal elements of a lower triangular square matrix A are all different from zero, then the matrix A will always be

A

symmetric

B

non-symmetric

C

singular

D

non-singular

Open complete paper

7

2007 · Engineering Sciences · Linear Algebra · Algebra of real matrices

Engineering Sciences (XE) 2007

The value of N is

A

4

B

5

C

6

D

7

Open complete paper

8

2007 · Engineering Sciences · Linear Algebra · Algebra of real matrices

Engineering Sciences (XE) 2007

The output of the above counter is given to the circuit shown below, which consists of 3 line to 8 line decoder and LEDs.
The LEDs that will never glow are

A

D₀ and D₇

B

D₀ and D₃

C

D₂ and D₃

D

D₅ and D₇

Open complete paper

9

2007 · Engineering Sciences · Linear Algebra · Algebra of real matrices

Engineering Sciences (XE) 2007

Let \( M \) be a \( 2 \times 2 \) matrix with eigenvalues \( 1 \) and \( 2 \). Then \( M^{-1} \) is

A

\(\frac{M - 3I}{2} \)

B

\(\frac{3I + M}{2} \)

C

\(\frac{3I - M}{2} \)

D

\(\frac{-M - 3I}{2} \)

Open complete paper

10

2007 · Engineering Sciences · Linear Algebra · Algebra of real matrices

Engineering Sciences (XE) 2007

The value of \( a_{11} \) is

A

0

B

1

C

2

D

3

Open complete paper

11

2007 · Engineering Sciences · Linear Algebra · Algebra of real matrices

Engineering Sciences (XE) 2007

Let S₁ be the sum of the eigen values of a 2 x 2 matrix P and S₂ be the sum of the eigen values of another 2 x 2 matrix Q. If S₁ = S₂, then P and Q are

A

\(\begin{pmatrix} 4 & 1 \\ 3 & 5 \end{pmatrix}\) and \(\begin{pmatrix} 1 & 4 \\ 2 & 3 \end{pmatrix}\)

B

\(\begin{pmatrix} 3 & 4 \\ 5 & 1 \end{pmatrix}\) and \(\begin{pmatrix} 2 & 4 \\ 3 & 1 \end{pmatrix}\)

C

\(\begin{pmatrix} 4 & 1 \\ 3 & 5 \end{pmatrix}\) and \(\begin{pmatrix} 3 & 4 \\ 1 & 5 \end{pmatrix}\)

D

\(\begin{pmatrix} 1 & 3 \\ 4 & 5 \end{pmatrix}\) and \(\begin{pmatrix} 4 & 3 \\ 1 & 2 \end{pmatrix}\)

Open complete paper

12

2007 · Engineering Sciences · Linear Algebra · Algebra of real matrices

Engineering Sciences (XE) 2007

If two eigen values of the matrix \( M = \begin{pmatrix} 2 & 6 & 0 \\ 1 & p & 0 \\ 0 & 0 & 3 \end{pmatrix} \) are \(-1\) and \(4\), then the value of \(p\) is

A

4

B

2

C

1

D

-1

Open complete paper

13

2008 · Engineering Sciences · Linear Algebra · Algebra of real matrices

Engineering Sciences (XE) 2008

If the characteristic equation of a \(3 \times 3\) matrix is \(\lambda^3 - \lambda^2 + \lambda - 1 = 0\), then the matrix should be

A

Hermitian

B

unitary

C

skew symmetric

D

identity

Open complete paper

14

2008 · Engineering Sciences · Linear Algebra · Algebra of real matrices

Engineering Sciences (XE) 2008

One of the eigenvalues of a \( 3 \times 3 \) matrix \( M \) is 3. If the determinant of the matrix \( M \) is 24 and the trace is 9, then the smallest eigenvalue of the matrix \( M^{-1} \) is

A

1/8

B

1/4

C

1/3

D

1/2

Open complete paper

15

2008 · Engineering Sciences · Linear Algebra · Algebra of real matrices

Engineering Sciences (XE) 2008

The system of equations
\(ax + by + a^2 = 0\)
\(bx + ay - b^2 = 0\)
\(x + y + a - b = 0\)

A

admits unique solution if \(a = b \neq 0\)

B

admits unique solution if \(a = -b \neq 0\)

C

admits unique solution if \(a = b = 0\)

D

does not admit unique solution

Open complete paper

16

2008 · Engineering Sciences · Linear Algebra · Algebra of real matrices

Engineering Sciences (XE) 2008

The matrix
\[ \begin{bmatrix} l & 0 & \sin \theta \\ 0 & 1 & m \\ n & 0 & \cos \theta \end{bmatrix} \]
is orthogonal, if

A

\(l = -\sin \theta, \; m = -\cos \theta, \; n = 0\)

B

\(l = -\sin \theta, \; m = 0, \; n = \cos \theta\)

C

\(l = \cos \theta, \; m = \sin \theta, \; n = 0\)

D

\(l = -\cos \theta, \; m = 0, \; n = \sin \theta\)

Open complete paper

17

2008 · Engineering Sciences · Linear Algebra · Algebra of real matrices

Engineering Sciences (XE) 2008

On solving the system of equations \( 4x + z = 5 \) \( x + 2y + 3z = 1 \) \( -y - 4z = 3, \) by \( LU \)- decomposition with \( u_{ii} = 1 \) for \( i = 1, 2, 3 \), the values of \( u_{23} \) and \( l_{31} \) are respectively

A

1.375 and -4.250

B

2.750 and -3.625

C

1.375 and -2.625

D

2.750 and -4.250

Open complete paper

18

2008 · Engineering Sciences · Linear Algebra · Algebra of real matrices

Engineering Sciences (XE) 2008

The eigenvalues of the matrix \[\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\] are

A

1,-1

B

i,i

C

1,i

D

i,-i

Open complete paper

19

2008 · Engineering Sciences · Linear Algebra · Algebra of real matrices

Engineering Sciences (XE) 2008

The lower triangular matrix \( L \) in the LU factorization of the matrix \[ \begin{pmatrix} 25 & 5 & 4 \\ 10 & 8 & 16 \\ 8 & 10 & 22 \end{pmatrix} \] is written as \( \begin{pmatrix} 1 & 0 & 0 \\ L_{21} & 1 & 0 \\ L_{31} & L_{32} & 1 \end{pmatrix} \). The element \( L_{32} \) is

A

1.0

B

1.4

C

0.4

D

0.32

Open complete paper

20

2009 · Engineering Sciences · Linear Algebra · Algebra of real matrices

Engineering Sciences (XE) 2009

Let A and B be two similar square matrices of order two. If 1 and -2 are the eigenvalues of A, then the Trace of B is

A

-2

B

-1

C

1

D

2

Open complete paper