# Matrix Algebra Gate Questions | Engineering Mathematics

 Question 1
The number of distinct eigenvalues of the matrix is equal to __________.
 A 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
 A -2.5 B 0 C 15 D 25
Question 2 Explanation:
If sum of all the rows or columns are same, then that
sum = one eigen value of that matrix

Sum of all rows
(15-x) * |Matrix| = 0
15 is a factor .

Another Approach Question 3
The rank of the matrix M = is
 A 0 B 1 C 2 D 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
 Question 4

Consider the following statement about the linear dependence of the real valued functions y1= 1, y2= x and y 3 =x2, over the field of real numbers.
I. y1, y2 and y3 are linearly independent on -1 x 0
II. y1 , y2 and y3 are linearly dependent on 0 x 1
III. y1 , y2 and y3 are linearly independent on 0 x 1
IV. y1, y2 and y3 are linearly dependent on -1 x 0
Which one among the following is correct?

 A Which one among the following is correct? B Both I and III are true C Both II and IV are true D Both III and IV are true
Question 4 Explanation:
Given
y1= 1, y 2= x, y 3= x2
Consider = =2 ≠0 y1, y2, y3 are linearly independent x
 Question 5
Consider matrix and vector The number of distinct real values of for which the equation has infinitely many solution is ___________.
 A Fill in the Blank Type Question
Question 5 Explanation: Question 6
Let M4 = I, (where I denotes the identity matrix) and M ≠ I, M2 ≠I and M3 ≠I. Then, for any natural number k, M-1  equals:
 A M4k+1 B M4k + 2 C M4k+3 D M4k
Question 6 Explanation: Question 7
The matrix has det A. = 100 and trace A=14
The value of is
 A 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 to
 A B C D Question 8 Explanation:
Given Product of eigen values = Det of A Question 9
For A = , the determinant of AT A–1 is
 A sec2 x B cos 4x C 1 D 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?
 A B (cM)T = c(M)T C D Question 10 Explanation:
Matrix multiplication is not commutative in general.
All other are properties of matrix. Question 11
A real (4 × 4) matrix A satisfies the equation A2 = I, where I is the (4 × 4) identity matrix. The positive eigen value of A is __________.
 A 1 B 2 C 4 D 8
Question 11 Explanation: Question 12
Consider the matrix: Which is obtained by reversing the order of the columns of the identity matrix I6.
Let P= I6 + αJ6 where α is a non-negative real number.
The value of α for which det(P) = 0 is ___________.
 A 0 B 1 C 2 D 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 ________.
 A 100 B 8 C 1000 D 200
Question 13 Explanation:
Determinant of matrix AB will be
Determinant of (A) * Determinant of (B)
5*40 = 200
 Question 14
The system of linear equations A a unique solution B infinitely many solutions C no solution D 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 ________.
 A 40 B 45 C 49 D 52
Question 15 Explanation: Question 16
Which one of the following statements is NOT true for a square matrix?
 A If A is upper triangular, the eigenvalues of A are the diagonal elements of it B If A is real symmetric, the eigenvalues of A are always real and positive C If A is real, the eigenvalues of A and AT are always the same D 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 A 2 B 5 C 8 D 16
Question 17 Explanation: Question 18
Given that and , the value of A3 is
 A 15 A + 12 I B 19A + 30 I C 17A + 15 I D 17 A + 21I
Question 18 Explanation:
Characteristic equation of A is  (-5-λ)(-λ) + 6 = 0 So, (by Cayley Hamilton theorem)
A2 = –5A – 6I
Multiplying by A on both sides, we have,
A3 =-5A2-6A
A3 = -5(-5A-6I)-6A
= 19A + 30I
There are 18 questions to complete.

### This Post Has 8 Comments

1. Anonymous

I cant solve the questioner! because its so hard!

1. Karthik

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

2. Anonymous

right answer of Question no. 9 is 1

1. Anonymous

no, its correct

3. 758542

Yes

4. Gate aspirant

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.

5. Gate aspirant

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.

6. Gatee

A transpose is not same as matrix A