Matrices Test

Result

Matrices Test - 6

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Question 1
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A
an invertible matrix

B
a nilpotent matrix

C
an idempotent matrix

D
none of these

Solution

A square matrix A for which Aⁿ = 0 , where n is a positive integer, is called a Nilpotent matrix.
Question 2
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A

B

C

D
none of these.

Solution
Question 3
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If A any square matrix then which of the following is not symmetric ?

A
A −A^t

B
A+A^t

C
AA^t

D
A^t A

Solution

For every square matrix (A –A ’) is always skew –symmetric.
Question 4
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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

A
ad –bc = ±1

B
ad –bc = - 1

C
ad –bc = 1

D
ad - bc need not be equal to ±1.

Solution

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

since the solution is unique in integers.
Question 5
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The system of equations, x + y + z = 1, 3 x + 6 y + z = 8, αx + 2 y + 3z = 1 has a unique solution for

A
αnot equal to 0

B
all integral α

C
all rational α

D
all real α

Solution

The given system of equations has unique solution , if

⇒1(18 −2)−1(9 −α) ⇒13 −5 α≠0 ⇒α≠13/5 + 1(6 −6 α) ≠0
Therefore , unique solution exists for all integral values of α.
Question 6
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If A and B are symmetric matrices of the same order, then

A
AB is symmetric matrix

B
AB –BA is a symmetric matrix

C
A –B is a skew-symmetric matrix

D
AB + BA is a symmetric matrix

Solution

If A and B are symmetric matrices of the same order, then , AB + BA is always a symmetric matrix.
Question 7
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If A = [a_ij ]_{2 ×2} where a_ij = i + j, then A is equal to

A

B

C

D

Solution

If A = [a_ij ]_2x2 where a_ij = i + j, then,
Question 8
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Each diagonal element of a skew-symmetric matrix is

A
negative.

B
Zero

C
non-real

D
Positive

Solution

The diagonal elements of a skew-symmetric is zero.
Question 9
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The system of equations, x + y + z = 6, x + 2 y + 3 z = 14, x + 3 y + 5z = 20 has

A
infinitely many solutions

B
a unique solution

C
only finitely many solutions

D
no solution.

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

A

B

C

D

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