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QUESTION: 1

If , then

Solution:

On operating [R_{1} → R_{1}-R_{2} and R_{3}→ R_{3}-R_{2}]

QUESTION: 2

Let , then the rank of M is equal to

Solution:

then

If

hence rank M = 2

QUESTION: 3

If , then

Solution:

Given

i.e., AA^{T }= I

A^{T} = A^{-1}

QUESTION: 4

If , then the matrix A is equal to

Solution:

we have

QUESTION: 5

If A is a non-zero column matrix of order m * 1 and B is a non-zero row matrix of order 1 * n, then rank of AB is equal to

Solution:

Let and B = [b_{11} b_{12} b_{13}..... b_{1n}] be two non-zero columns and row matrices respectively.

Since A, B are non-zero matrices.

∴ Matrix AB will be a non-zero matrix. The matrix AB will have atleast one non-zero element obtained by multiplying corresponding non-zero elements of A and B. All the two-rowed minors of AB clearly vanish.

∴ rank of AB = 1

QUESTION: 6

If then

Solution:

Applying C_{1} → C_{1} - C_{2} + C_{3} - C_{4}, we get

expanding with respect to first column

expanding with respect to the third column

= (a - 1)^{3} (a - 1)^{2} (a - 1) = (a - 1)^{6}.

QUESTION: 7

If a 3 x 3 matrix A has its inverse equal to A, then A^{2} is equal to , ......

Solution:

given

A^{-1 }= A

⇒ A^{2 }= I

QUESTION: 8

The sum of the eigen values of the matrix for real and negative values of x is

Solution:

Eigen values are given by the solution of equation

Since x is real and negative , put x = -k, where k is positive constant

∴ ( 3 - λ ) ( 1 - λ ) + 4 k = 0

⇒ λ^{2} - 4λ + 3 + 4 k = 0

If λ_{2} and λ_{2} be the solutions of the above equation, then λ_{2} and λ_{2 } are the eigen values.

Now,Sum of the eigen values = Sum of the roots of the above equation

i.e

QUESTION: 9

Let U be a 3 * 3 complex Hermitian matrix which is unitary. Then the distinct eigen values of U are

Solution:

By the well known facts, we know that a Hermitian matrix can have only real eigenvalues and a unitary matrix can have eigenvalues of unit modules. If U be 3 * 3 complex Hermitian matrix which is unitary also then distinct eigenvalues of U are of unit modulus and real. They can be

QUESTION: 10

Let the characteristics equation of a matrix M be λ^{2} - λ - 1 =0, then

Solution:

Given characteristic equation is λ^{2} - λ - 1 =0, By C - H theorem M^{2} - M - 1 = 0

⇒ M(M - 1) = i

⇒ M^{-1} = M - 1

QUESTION: 11

If then find eigen values of the matrix I + A + A^{2}, where I denotes the identity matrix

Solution:

A + I is a triangular matrix. We know that the e igen values of a triangular matrix are its diagonal elements. Eigen values of A^{2} + A + 2 are 3, 7, 13

QUESTION: 12

Let An eigenvalue of A is 2. Find a basis for the corresponding eigen space.

Solution:

Row reduce the augmented matrix for (A - 2i)x = 0.

At this point, we are confident that 2 is indeed an eigenvalue of A because the equation (A - 2l)x = 0 have free variables. The general solution is

The eigen space, is

QUESTION: 13

An arbitrary vector X is an eigen vector of the matrix

Solution:

(1,1) Since the matrix is triangular, the eigen values are 1, a, b. If (x_{1} x_{2}, x_{3}) is an arbitrary eigen vector, (say) corresponding to eigenvalue 1.

We have, x_{1} = x_{1, }ax_{2} = X_{2} which gives a = 1 and bx_{3} = x_{3} which gives b = 1. x_{2} & x_{3} being not zero,

∴ (a, b) = (1, 1).

QUESTION: 14

The values of λ and μ for which the equations x + y + z = 3, x + 3y + 2z = 6 and x + λy + 3z = μ have

Solution:

Given x + y + z = 3, x + 3y + 2z = 6 and x + λy + 3z = μ

If Δ ≠ 0 i.e. λ ≠ 0 the solution is unique.

Case 1 If λ ≠ 5 and μ s any real number, then unique solution exists

Case 2 If λ ≠ 5 ⇒ Δ = 0

If Δ_{1 }≠ 0 , μ ≠ 9 the system has no solution.

Case 3 If λ = 5 and μ = 9, infinite solutions exist

QUESTION: 15

If the matrix B is obtained from the matrix A by interchanging two rows, then-

Solution:

Interchanging two rows (or columns) change the sign of determinant.

QUESTION: 16

The rank of the Matrix is

Solution:

(Operating R_{1} -» R_{1} + R_{3} and R_{2} -> R_{2} - 2R_{3})

When

∴ ρ(A) = 1

where p(A) = number of non-zero rows

when

∴ ρ(A) = 2

when

∴ ρ(A) = 2

*Multiple options can be correct

QUESTION: 17

Consider the matrix, then

Solution:

, so eigen values are X = 1,1,4

Now co rresponding to λ = 1 .

(A - 1) x = 0

Clearly A has only one eigenvector corresponding to λ = 1.

⇒ A is not diagonalizable.

Characteristic eq^{n }will be,

*Multiple options can be correct

QUESTION: 18

Let V be the vector space of polynomials with degree not exceeding two. Let S = {1, x -1, x^{2} - 2x + 1} be a subset of V. Then which of the following(s) is/are true ?

Solution:

We know dim V = 3 and S is a linearly independent subset of V which has 3 elements = S form a basis for V

Now take (2x^{2}- 5x + 6) = a(1) + b ( x - 1) + c(x^{2} - 2x + 1) = 3(1) + (-1) ( x -1)+ 2(x^{2} - 2x + 1) = 2x^{2}- 5x + 6 = O ption (C) is correct

*Multiple options can be correct

QUESTION: 19

Let and be ordered basis of be a linear transformation such that

The matrix of T w.r. to basis β_{1} and β_{2} is then which of the followings is /are correct ?

Solution:

we have

*Multiple options can be correct

QUESTION: 20

If A be any matrix then AA^{T} and A^{T}A are

Solution:

Let A be real matrix , then take

(AA^{T})^{T} = (A^{T})A^{T} = AA^{T}

⇒ AA^{T} is Symmetric

now

(A^{T}A)^{T} = A^{T}(A^{T})^{T} = A^{T}A

⇒ A^{T}A is Symmetric

*Answer can only contain numeric values

QUESTION: 21

If matrix then the rank of the matrix A is _____________.

Solution:

Perform ing operations

we have

Performing Operation wee have

Hence all minors of 3rd and higher orders of the equivalent matrix are zero and one minor of order 2 whose value = 3 - 4 = -1 ≠0

Hence Rank of the equivalent matrix is Rank (A) = 2.

*Answer can only contain numeric values

QUESTION: 22

If A be an invertible matrix and Suppose that the inverse of 7 A is then the determinant of matrix A is ___________.

Solution:

Given

we know AA^{-1 }= 1

on Solving (1), (2), (3), (4) we get

*Answer can only contain numeric values

QUESTION: 23

If has the eigen values 3 and 9 then the sum o f the eigen values of A^{3} is __________.

Solution:

Characterstic eq^{n} of matrix A is

eigen values of matrix A is λ = 3, 9

then eigen values of A^{3} → 27 , 729

so = 27+729 = 756

*Answer can only contain numeric values

QUESTION: 24

The number of distinct real roots of in the interval is __________.

Solution:

Hence

or

or

(2 cos x + sin x) (sin x - cos x)^{2 }= 0

∴ tan x = - 2 , 1, But

*Answer can only contain numeric values

QUESTION: 25

The sum of distinct characteristic roots of the matrix is _______.

Solution:

The characteristic equation of A is

or (2 - λ) [(6 - λ)( 4 - λ) - 8 ] = 0

or ( 2 - λ) (λ^{2} - 10 λ+ 16) = 0

or ( 2 - λ) (λ - 2 ) ( λ - 8 ) = 0.

There fore the characteristic roots of A are given by λ = 2, 2, 8.

*Answer can only contain numeric values

QUESTION: 26

If 3, - 2 are the eigen values of a non-singular matrix A and |A| = 4, then the sum of the eigen values o f adj(A) are ________ .

Solution:

since and if λ is eigen value of A, then λ^{-1} is eigen value of A^{-1}.

Thus for adj (A)X = (A^{-1} X) |A| = |A| λ^{-1}

Thus, eigen value corresponding to λ = 3 is 4/3 and corresponding to λ = -2 is -4/2 = -2

Sum of eigen values = 4/3 - 2

= -2/3

= - 0.67

*Answer can only contain numeric values

QUESTION: 27

The system of linear equations

(4d - 1) x + y + z = 0, - y + z = 0, (4d - 1 )z = 0 has a non-trivial solution, if the value of d is ___ .

Solution:

The system of homogeneous linear equations has a non-trivial solution, if

*Answer can only contain numeric values

QUESTION: 28

Find the value of a for which the following system of equations over R is inconsistent.

Solution:

The augmented matrix is

Adding the first row to the second row, and 2 times the first row to the third row, we make the entries other than the pivot one in the 1 st column zeros. The row equivalent matrix not looks like

Next, we add 3 times the row to the third row to make the entry below the pivot one in the second column zero. This produces

which is the row echelon form of the augmented matrix. 1st last column is a pivot column unless a = 5. In other words, if the real number a * 5, then the given system is inconsistent.

*Answer can only contain numeric values

QUESTION: 29

If V is a vector space over an infinite field F such that dim V = 2, then the number of distinct subspaces V has ________.

Solution:

Here, V has a subspace of dimensions 0, 1 and 2 thus there are three subspaces.

*Answer can only contain numeric values

QUESTION: 30

The dimension of the subspace of R^{3} spanned by (-3,0, 1), (1, 2, 1) and (3, 0, - 1 ) is ______ .

Solution:

Here, we can se e that vectors (-3 , 0, 1) and (3, 0, - 1 ) are linearly dependent. Hence, dim = 2.

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