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QUESTION: 1

Solution:

A square matrix A for which A^{n}= 0 , where n is a positive integer, is called a Nilpotent matrix.

QUESTION: 2

Solution:

QUESTION: 3

If A any square matrix then which of the following is not symmetric ?

Solution:

For every square matrix (A – A’) is always skew – symmetric.

QUESTION: 4

Let a, b, c, d, u, v be integers. If the system of equations, a x + b y = u, c x + dy = v, has a unique solution in integers, then

Solution:

ax + by = u , cx +dy = v ,

since the solution is unique in integers.

QUESTION: 5

The system of equations, x + y + z = 1, 3 x + 6 y + z = 8, αx + 2 y + 3z = 1 has a unique solution for

Solution:

The given system of equations has unique solution , if

⇒1(18−2)−1(9−α) ⇒13−5α ≠ 0 ⇒ α ≠ 13/5 + 1(6−6α) ≠ 0

Therefore , unique solution exists for all integral values of α.

QUESTION: 6

If A and B are symmetric matrices of the same order, then

Solution:

If A and B are symmetric matrices of the same order, then , AB + BA is always a symmetric matrix.

QUESTION: 7

If A = [a_{ij}]_{2×2} where a_{ij}= i + j, then A is equal to

Solution:

If A = [a_{ij}]_{2x2} where a_{ij} = i + j, then,

QUESTION: 8

Each diagonal element of a skew-symmetric matrix is

Solution:

The diagonal elements of a skew-symmetric is zero.

QUESTION: 9

The system of equations, x + y + z = 6, x + 2 y + 3 z = 14, x + 3 y + 5z = 20 has

Solution:

The given system of equations does not has a solution if :

0 ⇒ 1(10 -9) - 1(5-3) + 1(3-2)

= 0 ⇒ 1-2 + 1 = 0

QUESTION: 10

The matrix of the transformation ‘reflection in the line x + y = 0 ‘ is

Solution:

Let x' and y' be the reflection of x and y, therefore :

Hence, reflection is on the line - x-y = 0⇒ x + y = 0

QUESTION: 11

A square matrix A = [a_{ij}]_{n×n} is called a diagonal matrix if a_{ij} = 0 for

Solution:

In a diagonal matrix all elements except diagonal elements are zero.i.e.

QUESTION: 12

If and then AB =

Solution:

QUESTION: 13

Solution:

or a symmetric matrix A’ = A . therefore ,

QUESTION: 14

If I_{n} is the identity matrix of order n, the (In)^{−1}

Solution:

Inverse of any identity matrix is always an identity matrix.

QUESTION: 15

If A = [x y z], and C = [xyt]^{t}, then ABC is

Solution:

QUESTION: 16

If a square matrix A has two identical rows or columns , then det.A is :

Solution:

Det.A = 0.

QUESTION: 17

For a skew symmetric odd ordered matrix A of integers, which of the following will hold true:

Solution:

Determinant of a skew symmetric odd ordered matrix A is always 0 .

QUESTION: 18

Matrix A when multiplied with Matrix C gives the Identity matrix I, what is C?

Solution:

Any square matrix when multiplied with its inverse gives the identity matrix. Note that non square matrices are not invertible.

QUESTION: 19

Let for any matrix M ,M^{−1}exist. Which of the following is not true.

Solution:

Clearly , (M−1)^{−1} = (M^{−1})^{1} is not true.

QUESTION: 20

Rank of a non-zero matrix is always

Solution:

Rank of a non zero matrix is always greater than or equal to 1.

QUESTION: 21

For a non-trivial solution | A | is

Solution:

An n×n homogeneous system of linear equations has a unique solution (the trivial solution) if and only if its determinant is non-zero. If this determinant is zero, then the system has an infinite number of solutions. i.e. For a non-trivial solution ∣A∣=0.

QUESTION: 22

If for a matrix A, A^{2}+I = O where I is the identity matrix, then A equals

Solution:

QUESTION: 23

Solution:

The given matrix is a skew – symmetric matrix.,therefore , A = - A’.

QUESTION: 24

The system of linear equations x + y + z = 2, 2x + y - z = 3, 3x + 2y - kz = 4 has a unique solution if ,

Solution:

The given system of equation has a unique solution if :

QUESTION: 25

The value of k for which the system of equations, x + k y + 3 z = 0, 3 x + k y – 2 z = 0, 2 x + 3 y – 4 z = 0, have a non-trival solution is

Solution:

The given system of equations has a non-trivial solution if :

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