Matrices Test

Result

Matrices Test - 7

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Question 1
1 / -0.25

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

A
I < j

B
I = j

C
I > j

D
I ≠ j

Solution

In a diagonal matrix all elements except diagonal elements are zero.i.e.
Question 2
1 / -0.25

If and then AB =

A
B²

B
0

C
A³

D
1

Solution
Question 3
1 / -0.25

A
a unit matrix

B
a skew- symmetric matrix.

C
a diagonal matrix

D
a symmetric matrix

Solution

or a symmetric matrix A ’= A . therefore ,
Question 4
1 / -0.25

If I_n is the identity matrix of order n, the (In)⁻¹

A
= O

B
n = I_n

C
= I_n

D
does not exist

Solution

Inverse of any identity matrix is always an identity matrix.
Question 5
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If A = [x y z], and C = [xyt]^t , then ABC is

A
1 ×1 matrix

B
3 ×3 matrix

C
not defined

D
none of these.

Solution

Matrix C is given in Transpose form. So, Matrix C is 3x1 order.
Question 6
1 / -0.25

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

A
0

B
-1

C
1

D
none of these.

Solution

Det.A = 0.
Question 7
1 / -0.25

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

A
det(A) = 9

B
det(A) = 81

C
det(A) = 0

D
et(A) = 4

Solution

Determinant of a skew symmetric odd ordered matrix A is always 0 .
Question 8
1 / -0.25

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

A
Identity matrix

B
Inverse of A

C
Square of A

D
Transpose of A

Solution

Any square matrix when multiplied with its inverse gives the identity matrix. Note that non square matrices are not invertible.
Question 9
1 / -0.25

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

A
(M⁻¹ )² = (M² )⁻¹

B
(M⁻¹ )⁻¹ = (M)

C
(M⁻¹ )⁻¹ = (M⁻¹ )¹

D
none of these.

Solution

Clearly , (M −1)⁻¹ = (M⁻¹ )¹ is not true.
Question 10
1 / -0.25

Rank of a non-zero matrix is always

A
⩾1

B
0

C
greater than 1

D
equal to 1

Solution

Rank of a non zero matrix is always greater than or equal to 1.
Question 11
1 / -0.25

For a non-trivial solution | A | is

A
|A| >0

B
|A| <0

C
|A| = 0

D
|A| ≠0

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 12
1 / -0.25

If for a matrix A, A² +I = O where I is the identity matrix, then A equals

A

B

C

D

Solution
Question 13
1 / -0.25

A
A = A ’

B
A = - A

C
A = - A ’

D
None of these.

Solution

The given matrix is a skew –symmetric matrix.,therefore , A = - A ’.
Question 14
1 / -0.25

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

A
k = 0

B
− 1 < k < 1

C
− 2 < k < 2

D
k ≠0

Solution

The given system of equation has a unique solution if :
Question 15
1 / -0.25

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

A
2/33

B
33

C
33/2

D
none of these

Solution

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