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Matrix MCQ - 3 - Mathematics MCQ

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30 Questions MCQ Test Topic-wise Tests & Solved Examples for Mathematics - Matrix MCQ - 3

Matrix MCQ - 3 for Mathematics 2024 is part of Topic-wise Tests & Solved Examples for Mathematics preparation. The Matrix MCQ - 3 questions and answers have been prepared according to the Mathematics exam syllabus.The Matrix MCQ - 3 MCQs are made for Mathematics 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Matrix MCQ - 3 below.

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Matrix MCQ - 3 - Question 1

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A matrix A such that A²= A, is called

A.
Idempotent
B.
Nilpotent
C.
Involutory
D.
Triangular

Matrix MCQ - 3 - Question 2

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Consider the matrix A = Then,

A.
the eigen values of A are ± 1
B.
sum of Alg. Mul. = 3
C.
sum of Geo. Mul. — 1
D.
None of these

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Matrix MCQ - 3 - Question 3

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A matrix A such that A² = I, is called

A.
Idempotent
B.
Involutory
C.
Nilpotent of order 2
D.
Triangular

Matrix MCQ - 3 - Question 4

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A matrix A such that A^m = 0 for some positive integer and A^k ≠ 0 for any k < m, is said to be

A.
Involutory
B.
Nilpotent of index m
C.
Nilpotent of orderk
D.
None of these

Matrix MCQ - 3 - Question 5

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A square matrix A such the A^T= —A, is called a

A.
Skew Symmetric matrix
B.
Symmetrix matrix
C.
Hermitian Matrix
D.
Skew Hermitian Matrix

Detailed Solution for Matrix MCQ - 3 - Question 5

Transpose of a matrix happens to be equal to the negative of itself, then one can say that the matrix is skew symmetric.

Matrix MCQ - 3 - Question 6

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A square matrix A such that A^T = -A, is called a

A.
Symmetric matrix
B.
Hermitian Matrix
C.
Skew Hermitian Matrix
D.
Skew Symmetric matrix

Detailed Solution for Matrix MCQ - 3 - Question 6

In mathematics, particularly in linear algebra, a skew-symmetric matrix is a square matrix whose transpose equals its negative.

Matrix MCQ - 3 - Question 7

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Let A and B two n × n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements.

I. rank(AB) = rank(A) rank(B)

II. det(AB) = det(A) det(B)

III. rank(A + B) ≤ rank(A) + rank(B)

IV. det(A + B) ≤ det(A) + det(B)

Which of the above statements are TRUE?

A.
I and II only
B.
I and IV only
C.
II and III only
D.
III and IV only

Detailed Solution for Matrix MCQ - 3 - Question 7

Example:
Consider two square matrices A and B each of order 2×2.

Statement II is TRUE.

Det(AB)= 5 = Det(A) * Det(B)

Statement III is TRUE.

Rank(A + B) = 2. Sum of rank of A and B is: 2 + 2 = 4. Therefore, the relation: Rank (A + B) ≤ Rank (A) + Rank (B) holds true

Therefore, the rank of the addition matrix is less than or equal to the sum of the rank of the individual matrices.

Matrix MCQ - 3 - Question 8

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A square matrix A is said to be Hermitian Matrix if

A.
A' = - A
B.
C.
A = A'
D.
None of these

Detailed Solution for Matrix MCQ - 3 - Question 8

Matrix MCQ - 3 - Question 9

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A square matrix A is said to be Skew Hermitian Matrix, if

A.
B.
A* = A
C.
D.
-A* = A

Detailed Solution for Matrix MCQ - 3 - Question 9

A matrix A = [a_ij] ∈ M_n is skew-Hermitian if A = − A ^*

Matrix MCQ - 3 - Question 10

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If A and B are two odd order Skew — symmetric matrices such that AB = BA, then what is the matrix AB?

A.
An orthogonal matrix
B.
A Skew — symmetric matrix
C.
A symmetric matrix
D.
An identity matrix

Matrix MCQ - 3 - Question 11

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Mark the incorrect statement

A.
The diagonal elements of a Skew symmetric matrix are all non — zero
B.
Every square matrix can be uniquely expressed as the sum of a symmetric and a Skew symmetric matrix
C.
If A is any square matrix, then A + A' is symmetric and A — A' is Skew symmetric
D.
If A is any matrix, then AA' and A'A are both symmetric

Matrix MCQ - 3 - Question 12

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The diagonal elements of a Skew Hermitian Matrix are

A.
All non — zero reals
B.
All purely imaginary number or zero
C.
Some non — zero reals and some imaginary
D.
None of these

Matrix MCQ - 3 - Question 13

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To convert a Hermitian Matrix into Skew Hermitian Matrix, the Hermitian Matrix must be multiplied by

A.
-1
B.
i
C.
-i
D.
None of these

Matrix MCQ - 3 - Question 14

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Which is not correct? If A is any square matrix, then... is Hermitian Matrix

A.
A + A*
B.
AA*
C.
A — A*
D.
A * A

Matrix MCQ - 3 - Question 15

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Every square matrix is uniquely expressible as

A.
The sum of a Hermitian Matrix and a Skew Hemitian matrix
B.
A + iB, where A and B both are symmetric
C.
A + iB, where A and B are real Skew symmetric matrices
D.
The sum of a symmetric and a Hermitian Matrices

Matrix MCQ - 3 - Question 16

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If A is any square matrix, then A — A^' is a

A.
Symmetric matrix
B.
Skew Hermitian matrix
C.
Hermitian Matrix
D.
Skew symmetric Matrix

Detailed Solution for Matrix MCQ - 3 - Question 16

(A − A')' = A' − (A')'

= A' − A

= −(A − A')
Therefore, it is a skew symmetric matrix

Matrix MCQ - 3 - Question 17

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If A and B are symmetric matrices of the same order, then which one of the following is not correct?

A.
A + B is a symmetric matrix
B.
B — BA is a symmetric matrix.
C.
AB + BA is a symmetric matrix
D.
A + A^T and B + B^T are symmetric matrices.

Matrix MCQ - 3 - Question 18

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A square matrix A is said to be ... if AA^T = I

A.
Idempotent
B.
Orthogonal
C.
Unitary
D.
Hermitian Matrix

Matrix MCQ - 3 - Question 19

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If A = satisfies the matrix equation A² — kA + 2I = 0, then what is the value of k?

A.
0
B.
1
C.
2
D.
3

Matrix MCQ - 3 - Question 20

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A square matirx A is said to be .... if A* A = I

A.
Orthogonal
B.
Unitary
C.
Symmetric
D.
Hermitian Matrix

Matrix MCQ - 3 - Question 21

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Under which one of the following condition does the system of equations have a unique solution?

A.
For all a ∈ R
B.
a = 8
C.
For all a ∈ Z
D.
a ≠ 6,8

Matrix MCQ - 3 - Question 22

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One of the integrating factor of the differential equation
(y² – 3xy)dx + (x²– xy)dy = 0 is

A.
1 / (x² y²)
B.
1 / (xy²)
C.
1 / -2x² y
D.
1/(xy)

Detailed Solution for Matrix MCQ - 3 - Question 22

(y² – 3xy)dx + (x² – xy) dy = 0;
M = y² – 3xy, N = x² – xy
Here differential equation is homogeneous, then
Mx + Ny = xy² – 3x²y + x²y – xy² = – 2x²y ≠ 0

Matrix MCQ - 3 - Question 23

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The product of two orthogonal matrices is a ... matrix

A.
Unitary
B.
Unit
C.
Orthogonal
D.
None of these

Matrix MCQ - 3 - Question 24

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The product of two Unitary matrices is a ________ matrix

A.
Unit
B.
Orthogonal
C.
Unitary
D.
None of the above

Detailed Solution for Matrix MCQ - 3 - Question 24

the product of two unitary matrices is always unitary.

Matrix MCQ - 3 - Question 25

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The points ( x_1,y₁ ) , ( x₂, y₂), ( x_3, y₃) are collinear if the rank of the matrix is

A.
3
B.
3 or 2
C.
2 or 1
D.
3 or 2 or 1

Matrix MCQ - 3 - Question 26

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A neccessary condition for the linear equations a₁x + b₁ = 0 and a₂x + b₂ = 0, to have a common solution is that

A.
a₁a₂ + b₁b₂ = 0
B.
a₁a₂ + b₁b₂ ≠ 0
C.
a₁b₂ — a₂b₁ = 0
D.
a₁b₂ + a₂b₁≠ 0

Detailed Solution for Matrix MCQ - 3 - Question 26

x = -b₁/a₁ and x = -b₂/a₂
-b₁/a₁ = -b₂/a₂
a₁xb₂ = a₂xb₁
a₁xb₂ - a₂xb₁ = 0

Matrix MCQ - 3 - Question 27

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A necesary and sufficient condition for the linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 to possess a unique solution is that

A.
a₁a₂ + b₁b₂ = 0
B.
a₁a₂ + b₁b₂ ≠ 0
C.
a₁b₂ — a₂b₁≠ 0
D.
a₁b₂ + a₂b₁ ≠ 0

Matrix MCQ - 3 - Question 28

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The value of the determinant is

A.
31
B.
-25
C.
19
D.
17

Matrix MCQ - 3 - Question 29

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Which of the following statement is incorrect?

A.
Transposition leaves the values of the determinant unchanged.
B.
If two rows of a determinant are proportional, the value of the determinat is zero
C.
If two rows of a determinant are interchanged, the value of the determinant so obtained is same as the value of the original determinant
D.
If the elements of any row of a determinant are multiplied by the same number k (say), then the value of the determinant so obtained is k times the value of the original determinant

Matrix MCQ - 3 - Question 30

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Let A and B be any two n x n matrices and tr(A) = Consider the following statement
I. tr(AB) = tr(BA)
II. tr(A + B) = tr. (A) + tr(B)
Which of the following statement given above is/are correct?

A.
I only
B.
II only
C.
Both I and II
D.
Neither I nor II

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