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This mock test of Test: Matrix for Mathematics helps you for every Mathematics entrance exam.
This contains 15 Multiple Choice Questions for Mathematics Test: Matrix (mcq) to study with solutions a complete question bank.
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QUESTION: 1

Let, and A^{–1} = Then (a + b) =

Solution:

We know AA^{-1} = I_{2}

QUESTION: 2

Given an orthogonal matrix A = [AA^{T}]^{-1} is

Solution:

We know

AA^{t} = I_{4}

[AA^{T}]^{-1} = [I_{4}]^{-1} = I_{4}

QUESTION: 3

The rank of the matrix is

Solution:

QUESTION: 4

The eigen values of a skew-symmetric matrix are

Solution:

**ANSWER :- c**

**Solution :- Let A be real skew symmetric and suppose λ∈C is an eigenvalue, with (complex) eigenvector v . Then, denoting by H hermitian transposition,**

**λvHv=vH(λv)=vH(Av)=vH(−AHv)=−(vHAH)v=−(Av)Hv=−(λv)Hv=−λ¯vHv **

**Since vHv≠0 , as v≠0 , we get**

**λ=−λ¯ **

**so λ is purely imaginary (which includes 0). Note that the proof works the same for a antihermitian (complex) matrix.**

**With a completely similar technique you can prove that the eigenvalues of a Hermitian matrix (which includes real symmetric) are real.**

QUESTION: 5

Rank of the matrix given below is:

Solution:

QUESTION: 6

The rank of a 3 x 3 matrix C (=AB), found by multiplying a non-zero column matrix A of size 3 x 1 and a non-zero row matrix B of size 1 x 3, is

Solution:

Then also every minor or order 2 is also zero.

∴ rank (C) = 1

QUESTION: 7

A is a 3 x 4 real matrix and A x = b is an inconsistent system of equations. The highest possible rank of A is

Solution:

Highest possible rank of A= 2 ,as Ax = b is an inconsistent system.

QUESTION: 8

Match the items in columns I and II.

Solution:

(P) Singular matrix → Determinant is zero

(Q) Non-square matrix → Determinant is not defined

(R) Real symmetric → Eigen values are always real

(S) Ortho gonal → Det erminant is always one

QUESTION: 9

[A] is a square matrix which is neither symmetric nor skew-symmetric and [A]^{T} is its transpose. The sum and difference of these matrices are defined as [S] = [A] + [A]^{T} and [D] = [A] -[A]^{T}, respectively. Which of the following statements is True?

Solution:

QUESTION: 10

Given matrix [A] = the rank of the matrix is

Solution:

∴ Rank (A) = 2

QUESTION: 11

Real matrices [A]_{3 x 1}, [B]_{3 x 3}, [C]_{3 x 5},[D]_{5 x 3}, [E]_{5 x 5} and [F]_{5 x 1} are given. Matrices [B] and [E] are symmetric.

Following statements are made with respect to these matrices.

1. Matrix product [F]^{T} [C]^{T} [B] [C] [F] is a scalar.

2. Matrix product [D]^{T} [F] [D] is always symmetric.

With reference to above statements, which of the following applies?

Solution:

QUESTION: 12

Consider the matrices X _{(4 × 3)}, Y _{(4 × 3)} and P _{(2 × 3)}. The order or P (X^{T}Y)^{–1}P^{T}] ^{T} will be

Solution:

When two matrices (x,y) and (a,b) are to be multiplied, the multiplication is possible

when y = a, and the resultant matrix is of the order (x * a)

X^{T} = The matrix in which rows and columns are interchanged

{(2 x 3) x [(3x4) x (4x3)]^{-1} x (3x2)}^{T }= { (2x3) x (3x3) x (3x2) }^{T} = { (2x3) x (3x2) }^{T} **= 2 x 2**

QUESTION: 13

The inverse of the 2 x 2 matrix is,

Solution:

QUESTION: 14

The product of matrices (PQ)^{-1} P is

Solution:

(PQ)^{-1}P = Q^{-1}P^{-1}P = Q^{-1}

QUESTION: 15

A square matrix B is skew-symmetric if

Solution:

B^{T} = – B

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