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This mock test of Test: Linear Algebra for GATE helps you for every GATE entrance exam.
This contains 20 Multiple Choice Questions for GATE Test: Linear Algebra (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

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: 4

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: 5

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: 6

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: 7

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

Solution:

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:

A must be invertible.

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:

Determinant of a **skew-symmetric even ordered matrix** A should be a perfect square.

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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