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QUESTION: 1

The rank of the following ( n + 1 ) x ( n + 1) matrix, where a is a real number is

Solution:

All the rows of the given matrix is same. So the matrix has only one independent row.

Rank of the matrix = Number of independent rows of the matrix.

∴ Rank of given matrix = 1

QUESTION: 2

Let AX - B be a system of linear equations where A is an m x n matrix and b is a m x 1 column vector and X is an x 1 column vector of unknown. Which of the following is false?

Solution:

Following are the possibilities for a system of linear equations:

(i) If matrix A and augmented matrix [AB] have same rank, then the system has solution otherwise there is no solution.

(ii) If matrix A and augmented matrix [AB] have same rank which is equal to the number of variables, then the system has unique solution and if B is zero vector then the system have only a trivial solution.

(iii) If matrix A and matrix [AB] have same rank which is less than the number of variables, then the system has infinite solution.

Therefore, option (c) is false because if m - n and B is non-zero vector, then it is not necessary that system has a unique solution, because m is the number of equations (quantity) and not the number of linearly independent equations (quality).

QUESTION: 3

The matrices commute under multiplication

Solution:

and

QUESTION: 4

Consider the following set of equations:

x + 2y = 5

4x + 8y = 12

3x + 6y + 3z = 15

This set

Solution:

Set of equations is

Above set of equations can be written as

Augmented matrix [AB] is given as

Performing gauss-Elimination on the above matrix

QUESTION: 5

An n x n array v is defined as follows:

The sum of the elements of the array v is

Solution:

The matrix V can be defined as

So above is antisymmetric matrix and the sum of the elements of any antisymmetric matrix is 0.

OR

Alternate Method:

QUESTION: 6

Consider the following statements:

S_{1}: The sum of two singular n x n matrices may be non-singular

S_{2}: The sum of two n x n non-singular matrices may be singular

Which of the following statements is correct?

Solution:

S_{1} is true

Consider two singular matrices

Sum of A and B is given as

However (A + B) is a non-singuiar matrix

So, S_{1} is true.

Now, consider two non-singuiar matrices

However (C + D) is a singular matrix. So, S_{2} is also true.

Therefore, both S_{1} and S_{2} are true.

QUESTION: 7

Let A, B, C, D be n x n matrices, each with non-zero determinant, If ABCD = I, then B^{-1} is

Solution:

A, B, C, D is n x n matrix.

Given, ABCD = I

QUESTION: 8

What values of x, y and z satisfy the following system of linear equations?

Solution:

QUESTION: 9

How many solutions does the following system of linear equations have?

- x + 5y = - 1

x - y = 2

x + 3y = 3

Solution:

The augmented matrix is

Using gauss-efimination on above matrix we get,

Rank [A | B] - 2 (number of non zero rows in [A | B])

Rank [A ] = 2 (number of non zero row s in [A ])

Rank [A | B] = Rank [A] = 2 = number of variables

∴ Unique solution exists.

QUESTION: 10

Consider the following system of equations in three real variables x_{1}, x_{2} and x_{3}

This system of equations has

Solution:

The augmented matrix for the given system is

Using gauss -elimination method on above matrix we get,

Since Rank ([A | B]) = Rank ([A]) = number of variables, the system has unique solution.

QUESTION: 11

What are the eigenvalues of the following 2 x 2 matrix?

Solution:

The characteristic equation of this matrix is given by

∴ The eigen values of A are 1 and 6.

QUESTION: 12

F is an n x n real matrix, b is an n x 1 real vector. Suppose there are two n x 1 vectors, u and v such that u ≠ v and Fu = b, Fv = b.

Which one of the following statements is false?

Solution:

Given that Fu= b and Fv= b

If F is non singular, then it has a unique inverse.

Now, u = F^{-1} b and v = F^{-1} b

Since F^{-1} is unique u = v but it is given that u ≠ v .

This is a contradiction. So F must be singular.

This means that

(a) Determinate of F is zero is true. Also

(b) There are infinite number of solution to Fx = b is true since |F| = 0

(c) There is an X ≠ 0 such that FX = 0 is also true, since X has infinite number of solutions, including the X = 0 solution.

(d) F must have 2 identical rows is false, since a determinant may become zero, even if two identical columns are present. It is not necessary that 2 identical rows must be present for |F| to become zero.

QUESTION: 13

What are the eigenvalues of the matrix P given below:

Solution:

Since P is a square matrix

Sum (eigen values) = Trace (P)

= a + a + a = 3a

Product of eigen values

Only choice (a) has sum of eigen values = 3a and product of eigen values = a^{3} - 2a.

QUESTION: 14

If the rank of a (5 x 6) matrix Q is 4, then which one of the following statements is correct?

Solution:

If rank of (5 x 6) matrix is 4, then surely it must have exactly 4 linearly independent rows as will as 4 linearly in dependent columns.

QUESTION: 15

The trace and determinant of a 2 x 2 matrix are known to be -2 and -35 respectively. It eigenvalues are

Solution:

Trace = Sum of principal diagonal element.

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