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This contains 20 Multiple Choice Questions for Railways Linear Algebra - MCQ Test (mcq) to study with solutions a complete question bank.
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QUESTION: 1

If A is a non–singular matrix and the eigen values of A are 2 , 3 , -3 then the eigen values of A^{-1} are

Solution:

If λ_{1} ,λ_{2} ,λ_{3} ....λ_{n }are the eigen values of a non–singular matrix A, then A^{-1} has the eigen values 1/λ_{1} ,1/λ_{2} ,1/λ_{3} ....1/λ_{n} Thus eigen values of A^{-1}are 1/2, 1/3, -1/3

QUESTION: 2

If -1 , 2 , 3 are the eigen values of a square matrix A then the eigen values of A^{2} are

Solution:

If λ_{1} ,λ_{2} ,λ_{3} ....λ_{n }are the eigen values of a matrix A, then A^{2} has the eigen values λ_{1}^{2} ,λ_{2}^{2} ,λ_{3}^{2} ....λ_{n}^{2} So, eigen values of A^{2 }are 1, 4, 9.

QUESTION: 3

If 2 , - 4 are the eigen values of a non–singular matrix A and |A| = 4, then the eigen values of adjA are

Solution:

QUESTION: 4

If 2 and 4 are the eigen values of A then the eigenvalues of A^{T} are

Solution:

Since, the eigenvalues of A and A^{T }are square so the eigenvalues of A^{T} are 2 and 4.

QUESTION: 5

If 1 and 3 are the eigenvalues of a square matrix A then A^{3} is equal to

Solution:

Since 1 and 3 are the eigenvalues of A so the characteristic equation of A is

Also, by Cayley–Hamilton theorem, every square matrix satisfies its own characteristic equation so

QUESTION: 6

If A is a square matrix of order 3 and |A| = 2 then A (adj A) is equal to

Solution:

QUESTION: 7

The sum of the eigenvalues of is equal to

Solution:

Since the sum of the eigenvalues of an n–square matrix is equal to the trace of the matrix (i.e. sum of the diagonal elements) so, required sum = 8 + 5 + 5 = 18

QUESTION: 8

If 1, 2 and 5 are the eigen values of the matrix A then |A| is equal to

Solution:

Since the product of the eigenvalues is equal to the determinant of the matrix so |A| = 1 x 2 x 5 = 10

QUESTION: 9

If the product of matrices

is a null matrix, then θ and Ø differ by

Solution:

QUESTION: 10

If A and B are two matrices such that A + B and AB are both defined, then A and B are

Solution:

Since A + B is defined, A and B are matrices of the same type, say m x n. Also, AB is defined. So, the number of columns in A must be equal to the number of rows in B i.e. n = m. Hence, A and B are square matrices of the same order.

QUESTION: 11

If A is a 3-rowed square matrix such that |A| = 2, then |adj(adj A^{2})| is equal to

Solution:

QUESTION: 12

then the value of x is

Solution:

QUESTION: 13

Solution:

Inverse matrix is defined for square matrix only.

QUESTION: 14

Solution:

QUESTION: 15

A set of linear equations is represented by the matrix equation Ax = b. The necessary condition for the existence of a solution for this system is

Solution:

QUESTION: 16

Select a suitable figure from the four alternatives that would complete the figure matrix.

Solution:

In each row (as well as each column), the third figure is a combination of all the elements of the first and the second figures

QUESTION: 17

For a skew symmetric even ordered matrix A of integers, which of the following will not hold true:

Solution:

QUESTION: 18

Solution:

QUESTION: 19

Matrix D is an orthogonal matrix The value of B is

Solution:

For orthogonal matrix

QUESTION: 20

Solution:

From linear algebra for A_{nxn} triangular matrix . DetA = The product of the diagonal entries of A

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