Linear Algebra - Engineering Mathematics - Metallurgical Engineering Previous Year Questions

1

2007 · Metallurgical Engineering · Engineering Mathematics · Linear Algebra

Metallurgical Engineering (MT) 2007

The determinant of the matrix \(\begin{bmatrix} 1 & 3 & 2 \\ 2 & 6 & 4 \\ -5 & 3 & 1 \end{bmatrix}\) is

A

-10

B

-5

C

0

D

5

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2

2008 · Metallurgical Engineering · Engineering Mathematics · Linear Algebra

Metallurgical Engineering (MT) 2008

Determinant of \(\begin{vmatrix} 3 & 1 & 2 \\ 1 & 2 & 1 \\ 4 & 2 & 3 \end{vmatrix}\) is

A

-2

B

-1

C

1

D

2

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3

2009 · Metallurgical Engineering · Engineering Mathematics · Linear Algebra

Metallurgical Engineering (MT) 2009

In an \( n \times n \) identity matrix, the trace equals

A

0

B

1

C

n

D

\(n^2 \)

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4

2010 · Metallurgical Engineering · Engineering Mathematics · Linear Algebra

Metallurgical Engineering (MT) 2010

Which of the following is NOT a property of a n × n singular matrix?

A

Rank = n

B

Linearly dependent row vectors

C

Zero diagonal in Gauss elimination

D

Linearly dependent column vectors

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5

2012 · Metallurgical Engineering · Engineering Mathematics · Linear Algebra

Metallurgical Engineering (MT) 2012

A is a 2 × 2 matrix with det A = 2. The det (2A) is

A

4

B

8

C

32

D

16

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6

2012 · Metallurgical Engineering · Engineering Mathematics · Linear Algebra

Metallurgical Engineering (MT) 2012

A is a 2 × 2 matrix given below: \[ \mathbf{A} = \begin{pmatrix} -3 & 1 \\ -1 & -1 \end{pmatrix} \] The eigenvalues of A are

A

-2, -2

B

-3, -1

C

2, 2

D

3, 1

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7

2012 · Metallurgical Engineering · Engineering Mathematics · Linear Algebra

Metallurgical Engineering (MT) 2012

Raju has 14 currency notes in his pocket consisting of only Rs. 20 notes and Rs. 10 notes. The total money value of the notes is Rs. 230. The number of Rs. 10 notes that Raju has is

A

5

B

6

C

9

D

10

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8

2013 · Metallurgical Engineering · Engineering Mathematics · Linear Algebra

Metallurgical Engineering (MT) 2013

Which one of the following attributes is NOT correct for the matrix?
\[ \begin{bmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix} \] , where \( \theta = 60^\circ \)

A

orthogonal

B

singular

C

skew-symmetric

D

positive-definite

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9

2013 · Metallurgical Engineering · Engineering Mathematics · Linear Algebra

Metallurgical Engineering (MT) 2013

X and Y are two positive real numbers such that 2X + Y ≤ 6 and X + 2Y ≤ 8. For which of the following values of (X, Y) the function f(X, Y) = 3X + 6Y will give maximum value?

A

(4/3, 10/3)

B

(8/3, 20/3)

C

(8/3, 10/3)

D

(4/3, 20/3)

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10

2014 · Metallurgical Engineering · Engineering Mathematics · Linear Algebra

Metallurgical Engineering (MT) 2014

If all the elements of one row of a 3 × 3 matrix are multiplied by 3, the determinant of the matrix changes by a factor of ____.

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11

2016 · Metallurgical Engineering · Engineering Mathematics · Linear Algebra

Metallurgical Engineering (MT) 2016

The sum of the digits of a two digit number is 12. If the new number formed by reversing the digits is greater than the original number by 54, find the original number.

A

39

B

57

C

66

D

93

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12

2017 · Metallurgical Engineering · Engineering Mathematics · Linear Algebra

Metallurgical Engineering (MT) 2017

For the matrix. \(A = \begin{bmatrix} 1 & 1 & 2 \\ 2 & 1 & 1 \\ 1 & 1 & 2 \end{bmatrix}\), \(AA^T\) is

A

\(\begin{bmatrix} 6 & 5 & 6 \\ 5 & 6 & 6 \\ 6 & 5 & 6 \end{bmatrix}\)

B

\(\begin{bmatrix} 6 & 5 & 6 \\ 5 & 6 & 6 \\ 5 & 5 & 6 \end{bmatrix}\)

C

\(\begin{bmatrix} 6 & 5 & 6 \\ 5 & 6 & 5 \\ 6 & 6 & 6 \end{bmatrix}\)

D

\(\begin{bmatrix} 6 & 5 & 6 \\ 5 & 6 & 5 \\ 6 & 5 & 6 \end{bmatrix}\)

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13

2019 · Metallurgical Engineering · Engineering Mathematics · Linear Algebra

Metallurgical Engineering (MT) 2019

One of the eigenvalues for the following matrix is ______. \[ \begin{bmatrix} a & 2 \\ 8 & a \end{bmatrix} \]

A

\(a-4\)

B

\(-a-4\)

C

4

D

\(-4\)

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14

2020 · Metallurgical Engineering · Engineering Mathematics · Linear Algebra

Metallurgical Engineering (MT) 2020

M and N are 3 × 3 matrices. If the det(M) is –9 and the det(N) is –14, then the det(NM) is ______ (round off to the nearest integer).

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15

2021 · Metallurgical Engineering · Engineering Mathematics · Linear Algebra

Metallurgical Engineering (MT) 2021

For the matrix given below, the eigenvalues are: \[ \begin{pmatrix} 1 & 0 & -1 \\ 0 & 1 & 0 \\ -1 & 0 & 1 \end{pmatrix} \]

A

0, 2, 2

B

1, 1, 2

C

0, 1, 2

D

0, 1, 3

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16

2022 · Metallurgical Engineering · Engineering Mathematics · Linear Algebra

Metallurgical Engineering (MT) 2022

For a 3×3 matrix, the value of the determinant is −48 and the trace is 8. If one of the eigenvalues is 4, the other two are ______

A

2, −3

B

1, −3

C

6, −2

D

−4, 0

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17

2023 · Metallurgical Engineering · Engineering Mathematics · Linear Algebra

Metallurgical Engineering (MT) 2023

The sum of eigen values of the matrix \[ \begin{bmatrix} 4 & 3 & 2 \\ 0 & -1 & 2 \\ 0 & 0 & -3 \end{bmatrix} \] is __________ (in integer).

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18

2023 · Metallurgical Engineering · Engineering Mathematics · Linear Algebra

Metallurgical Engineering (MT) 2023

For the equation \[ \begin{vmatrix} x+3 & 3x+4 & 4x+5 \\ -2 & -3 & -4 \\ -3 & -4 & -5 \end{vmatrix} = 0 \] the value of x is __________ (in integer).

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19

2024 · Metallurgical Engineering · Engineering Mathematics · Linear Algebra

Metallurgical Engineering (MT) 2024

Which of the following statements is/are correct for a square matrix A with real number entries?
A^T denotes the transpose of A and A⁻¹ denotes the inverse of A.

A

A is symmetric if A^T = −A.

B

A is skew-symmetric if A^T = −A.

C

If A is orthogonal, then A^T = A⁻¹.

D

If A is orthogonal, then its determinant is zero.

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20

2024 · Metallurgical Engineering · Engineering Mathematics · Linear Algebra

Metallurgical Engineering (MT) 2024

If $\begin{bmatrix} 1 & 2 \\ 8 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \lambda \begin{bmatrix} x \\ y \end{bmatrix}$, where $x, y$ are not identically zero, then the values of $\lambda$ are

A

$5, -3$

B

$4, -4$

C

$3, -5$

D

$5, -4$

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