Question 1 |

The number of distinct eigenvalues of the matrix is equal to __________.

Fill in the Blank Type Question |

Question 2 |

Consider the 5 × 5 matrix It is given that A has only one real Eigen value. Then the real Eigen value of A is

-2.5 | |

0 | |

15 | |

25 |

Question 2 Explanation:

**If sum of all the rows or columns are same, then that**

sum = one eigen value of that matrix

sum = one eigen value of that matrix

Sum of all rows

**(15-x) * |Matrix| = 0**

15 is a factor .

15 is a factor .

**Another Approach**Question 3 |

The rank of the matrix M = is

0 | |

1 | |

2 | |

3 |

Question 3 Explanation:

|M |=

= 5(0-12) – 10(6-6) +10(6-0)= -60-0+60=0

But a 2×2 minor, =0-10=-10≠0

Rank = 2

= 5(0-12) – 10(6-6) +10(6-0)= -60-0+60=0

But a 2×2 minor, =0-10=-10≠0

Rank = 2

Question 4 |

Consider the following statement about the linear dependence of the real valued functions y_{1}= 1, y_{2}= x and y _{3} =x^{2}, over the field of real numbers.

I. y_{1}, y_{2} and y_{3} are linearly independent on -1 ≤ x ≤ 0

II. y_{1} , y_{2} and y_{3} are linearly dependent on 0 ≤ x ≤1

III. y_{1} , y_{2} and y_{3} are linearly independent on 0 ≤x ≤1

IV. y_{1}, y_{2} and y_{3} are linearly dependent on -1 ≤ x ≤ 0

Which one among the following is correct?

Which one among the following is correct? | |

Both I and III are true | |

Both II and IV are true | |

Both III and IV are true |

Question 5 |

Consider matrix and vector The number of distinct real values of for which the equation has infinitely many solution is ___________.

Fill in the Blank Type Question |

Question 5 Explanation:

Question 6 |

Let M

^{4}= I, (where I denotes the identity matrix) and M ≠ I, M^{2}≠I and M^{3}≠I. Then, for any natural number k, M^{-1 }equals:M ^{4k+1} | |

M ^{4k + 2} | |

M ^{4k+3} | |

M ^{4k} |

Question 6 Explanation:

Question 7 |

The matrix has det A. = 100 and trace A=14

The value of is

The value of is

Fill in the Blank Type Question |

Question 7 Explanation:

Question 8 |

Consider a square matrix Where

*x*is unknown. If the eigenvalues of the matrix A are and the x is equal toQuestion 8 Explanation:

Given

Product of eigen values = Det of A

Product of eigen values = Det of A

Question 9 |

For A = , the determinant of A

^{T}A^{–1}issec ^{2} x | |

cos 4x | |

1 | |

0 |

Question 9 Explanation:

Question 10 |

For matrices of same dimension M, N and scalar c, which one of these properties DOES NOT ALWAYS hold?

(cM) ^{T} = c(M)^{T} | |

Question 10 Explanation:

Matrix multiplication is not commutative in general.

All other are properties of matrix.

Refer: www.khanacademy.org

All other are properties of matrix.

Refer: www.khanacademy.org

Question 11 |

A real (4 × 4) matrix A satisfies the equation A

^{2}= I, where I is the (4 × 4) identity matrix. The positive eigen value of A is __________.1 | |

2 | |

4 | |

8 |

Question 11 Explanation:

Question 12 |

Consider the matrix:

Which is obtained by reversing the order of the columns of the identity matrix I

Let

The value of

Which is obtained by reversing the order of the columns of the identity matrix I

_{6}.Let

**P= I**where_{6}+ αJ_{6}**α**is a non-negative real number.The value of

**α**for which**det(P) = 0**is ___________.0 | |

1 | |

2 | |

4 |

Question 12 Explanation:

Since, α is non-negative real number. Hence,

α = 1

Question 13 |

The determinant of matrix A is 5 and the determinant of matrix B is 40. The determinant of matrix AB is ________.

100 | |

8 | |

1000 | |

200 |

Question 13 Explanation:

Determinant of matrix AB will be

Determinant of (A)

5*40 = 200

so answer = 200

Determinant of (A)

*****Determinant of (B)5*40 = 200

so answer = 200

Question 14 |

The system of linear equations

a unique solution | |

infinitely many solutions | |

no solution | |

exactly two solutions |

Question 14 Explanation:

Clearly rank(A)=2 , rank(A/B) = 2 , number of unknowns = 3

so rank (A) = rank (A / B) =2

Since, rank (A) = rank (A / B) < number of unknowns

∴ Equations have infinitely many solutions.

Question 15 |

The maximum value of the determinant among all 2×2 real symmetric matrices with trace 14 is ________.

40 | |

45 | |

49 | |

52 |

Question 15 Explanation:

Question 16 |

Which one of the following statements is NOT true for a square matrix?

If A is upper triangular, the eigenvalues of A are the diagonal elements of it | |

If A is real symmetric, the eigenvalues of A are always real and positive | |

If A is real, the eigenvalues of A and A ^{T} are always the same | |

If all the principal minors of A are positive, all the eigenvalues of A are also positive |

Question 16 Explanation:

Question 17 |

Let A be an m × n matric and B an n × m matric. It is given that determinant determinant where is the k × k identity matrix. Using the above property, the determinant of the matrix given below is

2 | |

5 | |

8 | |

16 |

Question 17 Explanation:

Question 18 |

Given that and , the value of A

^{3}is15 A + 12 I | |

19A + 30 I | |

17A + 15 I | |

17 A + 21I |

Question 18 Explanation:

Characteristic equation of A is

So,

(by Cayley Hamilton theorem)

⇒ A

Multiplying by A on both sides, we have,

A

⇒ A

= 19A + 30I

_{(-5-λ)(-λ) + 6 = 0}So,

(by Cayley Hamilton theorem)

⇒ A

^{2}= –5A – 6IMultiplying by A on both sides, we have,

A

^{3}=-5A^{2}-6A⇒ A

^{3}= -5(-5A-6I)-6A= 19A + 30I

There are 18 questions to complete.

I cant solve the questioner! because its so hard!

First you need to try….nothing is impossible when you want fulfil your dream .. All d best

right answer of Question no. 9 is 1

no, its correct

Yes

Answer for question no 9 is option C that is 1. A inverse can be written an 1/A and A transpose is same as A ….so A*(1/A) gets cancelled and the answer is 1.

Answer for question no 9 is option C that is 1. A inverse can be written an 1/A and A transpose is same as A ….so A*(1/A) gets cancelled and the answer is 1.

A transpose is not same as matrix A