Test: Linear Algebra- 2

# Test: Linear Algebra- 2

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## 30 Questions MCQ Test GATE Computer Science Engineering(CSE) 2023 Mock Test Series | Test: Linear Algebra- 2

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Test: Linear Algebra- 2 - Question 1

### 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

Test: Linear Algebra- 2 - Question 2

### How many of the following matrices have an eigenvalue 1?

Detailed Solution for Test: Linear Algebra- 2 - Question 2

Characteristic equation is

(1)

*Multiple options can be correct
Test: Linear Algebra- 2 - Question 3

### In questions 1.1 to 1.7 below, one or more of the alternatives are correct. Write the code letter(s) a, b, c, d corresponding to the correct alternative(s) in the answer book.  Marks will be given only if all the correct alternatives have been selected and no incorrect alternative is picked up. 1.1). The eigen vector (s) of the matrix is (are)

Detailed Solution for Test: Linear Algebra- 2 - Question 3

Eigen values are: 0,0,0
The eigen vector should satisfy the equation: αz = 0

Test: Linear Algebra- 2 - Question 4

What are the eigenvalues of the following  matrix?

Detailed Solution for Test: Linear Algebra- 2 - Question 4

Let the eigen values be a,b
Sum of Eigen Values = Trace(Diagonal Sum)
⇒ a+b = 2+5 = 7
Product of Eigen Values = Det(A)
⇒ a.b = 6
Solving these we  get 1 and 6.. So, Option(B) is Correct ..

Test: Linear Algebra- 2 - Question 5

Consider the following matrix

If the eigenvalues of A are 4 and 8, then

Detailed Solution for Test: Linear Algebra- 2 - Question 5

Sum of eighen values is equal to trace(sum of diagonal elements),and product of eighen values is equal to det of matrix
So 2+y=8+4 y =10
2y-3x=32
Solving this we get x =-4
Option d is ans

Test: Linear Algebra- 2 - Question 6

a. Obtain the eigen values of the matrix

Detailed Solution for Test: Linear Algebra- 2 - Question 6

5(a) the eigen value for upper triangular/lower triangular/diagonal matrices are the diagonal elements of the matrix

Test: Linear Algebra- 2 - Question 7

Consider the following determinant

Which of the following is a factor of Δ?

Detailed Solution for Test: Linear Algebra- 2 - Question 7

R2->R2 - R1
R3 -> R3 - R2
you will gt det = (a-b)*(a-c)*(b+c)
in matrix operations, you cannot multiply rows or columns. That will not yield the same matrix. So abc is not correct

Test: Linear Algebra- 2 - Question 8

Let A be a matrix such that Ak = 0. What is the inverse of I - A?

Detailed Solution for Test: Linear Algebra- 2 - Question 8

Given AK = 0
I  -  Ak = I
I  -  Ak  = (I - A) (I + A + A2 + A3  +.........+Ak-1)
I  = (I - A) (I + A + A2 + A3  +.........+Ak-1)
(I - A)-1 =  (I + A + A2 + A3  +.........+Ak-1)

Test: Linear Algebra- 2 - Question 9

F is an n*n real matrix. b is an n*1 real vector. Suppose there are two n*1 vectors, u and v such that, u ≠ v and Fu = b, Fv = b. Which one of the following statements is false?

Detailed Solution for Test: Linear Algebra- 2 - Question 9

(A) : Correct. We are given

Since  so we have a non-zero solution  to homogeneous equation  Now any vector  is also a solution of and so we have infinitely many solutions of and so determinant of F is zero.
(B) : Correct. Consider a vector

So there are infinitely many vectors of the form   which are solutions to equation Fx = b.
(C) : Correct. In option (a), we proved that vector
(D) : False. This is not necessary.
So option (D) is the answer.

*Answer can only contain numeric values
Test: Linear Algebra- 2 - Question 10

Perform the following operations on the matrix

i. Add the third row to the second row
ii. Subtract the third column from the first column.
The determinant of the resultant matrix is _____.

Detailed Solution for Test: Linear Algebra- 2 - Question 10

Answer :- 0, because it is easy to see that first column and third column are multiple of each other.
Third column = First column * 15.
So rank is  < 3, so Determinant must be 0.
It stays zero as row & column transformations don't affect determinant.

*Answer can only contain numeric values
Test: Linear Algebra- 2 - Question 11

In the LU decomposition of the matrix , if the diagonal elements of U are both , then the lower diagonal entry of L is_________________.

Detailed Solution for Test: Linear Algebra- 2 - Question 11

Given

We need to find l21 and u12 l21 *  1 + l22 * 0  = 4 l21 = 4

l11 * U12 + 0 * 1 = 2 l11 = 2 U12 = 1 Putting value of l21 and u12 in (1), we get 4 * 1 + l22 * 1 = 9 l22 = 5

Test: Linear Algebra- 2 - Question 12

Let the characteristic equation of matrix  M be λ2 - λ - 1 = 0 . Then

Detailed Solution for Test: Linear Algebra- 2 - Question 12

I think we can solve using Cayley- Hamilton Theorem

Test: Linear Algebra- 2 - Question 13

Consider the following statements:

• SI: The sum of two singularn n x n matrices may be non-singular
• S2: The sum of two n x n non-singular matrices may be singular

Which one of the following statements is correct?

Detailed Solution for Test: Linear Algebra- 2 - Question 13

Yes A is correct option!, Both statements are True

Test: Linear Algebra- 2 - Question 14

How many 4 x 4 matrices with entries from  have odd determinant?
Hint: Use modulo  arithmetic.

Detailed Solution for Test: Linear Algebra- 2 - Question 14

whenever 1st row is 0 then its determent is 0 , and similarly if any 2 or more rows are linearly dependent then its |det|=0 In order to find the odd determinant the
1st row must be non zero --> totally(2^4-1) possibilities |0/1  0/1  0/1 0/1| like totally=16-1
2nd row must be non zero and not linearly depends on 1st row so--> totally (2^4-2) possibilities
for 3rd row it must be non-zero as well as not linearly depends on first 2 rows(not start with 0) --->totally (2^4-4)
for 4th row -->(2^4-8) :: total possibilities=(2^4-2^0) * (2^4-2^1) * (2^4-2^2) *(2^4-2^3)=15*14*12*8=20160 possible

Test: Linear Algebra- 2 - Question 15

The rank of the matrix

Detailed Solution for Test: Linear Algebra- 2 - Question 15

Rank of this matrix is 1 as the determint of 2nd order matrix is 0 and 1st order matrix is non zero so rank is 1

Test: Linear Algebra- 2 - Question 16

Let A be an n × n matrix of the following form.

What is the value of the determinant of A?

Detailed Solution for Test: Linear Algebra- 2 - Question 16

Best part of this question  is dont solve by mathematical procedures. Verification is very easy in this question. just put n=1 , u ll get a matrix like [3].. find itdeterminant.. determinant = 3. now check options.
by putting n=1, i am getting following results..

A. 5
B. 7
C. 3
D. 3

now check for n=2.
determinantdeterminant = 9-1 =8
put n=2 in C,D.
C = 7
D = 8

Test: Linear Algebra- 2 - Question 17

If matrix  and X2 - X + I = 0 (I is the identity matrix and  is the zero matrix), then the inverse of X is

Detailed Solution for Test: Linear Algebra- 2 - Question 17

Given, X2 - X + I  = O
=>   X2 = X - I
=> X-1 (X2)  = X-1(X - I)    {Multiplying X-1 both sides..}
=> X = I - X-1
=> X-1 = I - X

Which gives Option (B)..

Test: Linear Algebra- 2 - Question 18

The number of different  symmetric matrices with each element being either 0 or 1 is: (Note: (2,X) is same as 2X)

Detailed Solution for Test: Linear Algebra- 2 - Question 18

In symmetric matrix, So, we have choice only for either the upper triangular elements or the lower triangular elements. Number of such elements will be  Now, each element being either 0 or 1 means, we have 2 choices for each element and thus for  elements we have possibilities.

Test: Linear Algebra- 2 - Question 19

The rank of the following matrix, where  is a real number is

Detailed Solution for Test: Linear Algebra- 2 - Question 19

we can eliminate all other rows using row 1. in the last only 1 row will be left.

rank = no of non zero rows = 1

Test: Linear Algebra- 2 - Question 20

The rank of the matrix given below is:

Test: Linear Algebra- 2 - Question 21

Let A and B be real symmetric matrices of size. Then which one of the following is true?

Detailed Solution for Test: Linear Algebra- 2 - Question 21

Given A = A' and B = B'
(AB)' = B'A' = BA

Test: Linear Algebra- 2 - Question 22

The rank of matrix

Detailed Solution for Test: Linear Algebra- 2 - Question 22

Determinant comes out to be 0. So, rank cannot be 3. The minor

Test: Linear Algebra- 2 - Question 23

Let  A, B, C,D be  n * n  matrices, each with non-zero determinant. If ABCD = I, then B-1 is

Detailed Solution for Test: Linear Algebra- 2 - Question 23

Test: Linear Algebra- 2 - Question 24

In an M × N  matrix all non-zero entries are covered in  rows and  columns. Then the maximum number of non-zero entries, such that no two are on the same row or column, is

Detailed Solution for Test: Linear Algebra- 2 - Question 24

maximum number of non-zero entries, such that no two are on the same row or column

Any entry will be a member of some row and some column. So, with  rows we can have maximum  elements such that no row has a repeated element. Same is applicable for b columns also. So, combining both, answer should be .
We can also apply pigeonhole principle here. Let
P = min(a.b) be the number of holes. So, we can place up to
P non-zero entries (pigeons) and as soon as
(P + 1)th entry comes it must be making two entries in some column or row.

Test: Linear Algebra- 2 - Question 25

The matrices  commute under multiplication

Detailed Solution for Test: Linear Algebra- 2 - Question 25

The multiplication will commute if

Test: Linear Algebra- 2 - Question 26

If M is a square matrix with a zero determinant, which of the following assertion (s) is (are) correct?
S1: Each row of M can be represented as a linear combination of the other rows
S2: Each column of M can be represented as a linear combination of the other columns
S3: MX = 0 has a nontrivial solution
S4: M has an inverse

Detailed Solution for Test: Linear Algebra- 2 - Question 26

Since M has zero determinant, its rank is not full i.e. if M is of size 3*3, then its rank is not 3. So there is a linear combination of rows which evaluates to 0 i.e.

and there is a linear combination of columns which evaluates to 0 i.e.

Now any row Ri can be written as linear combination of other rows as :

Similar is the case for columns.
Now MX = 0 always has one solution : X = 0 (which is called trivial solution). Now if |M| = 0, then MX = 0 has non-trivial solutions also.
So (S1), (S2), and (S3) are true. So option D is correct.

Test: Linear Algebra- 2 - Question 27

Consider the following system of linear equations

Notice that the second and the third columns of the coefficient matrix are linearly dependent. For how many values of , does this system of equations have infinitely many solutions?

Detailed Solution for Test: Linear Algebra- 2 - Question 27

Determinant=0. Therefore apply reduction method on (A|B)

obtain the resultant matrix

or infinitely many solutions, we must have 2+1.5a=0 i.e., a=-4/3 so for only 1 value of a, this system has infinitely many solutions. So option (B) is correct.

Test: Linear Algebra- 2 - Question 28

Consider the following system of linear equations :

The system of equations has

Detailed Solution for Test: Linear Algebra- 2 - Question 28

rank of matrix = rank of augmented matrix = no of unknown = 3 so unique solution..

Test: Linear Algebra- 2 - Question 29

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

Detailed Solution for Test: Linear Algebra- 2 - Question 29

rank = r(A) = r(A|B) = 2
rank = total number of variables Hence, unique solution

Test: Linear Algebra- 2 - Question 30

Consider the following set of equations x + 2y = 54 x+8y = 123x + 6y+3z = 15. This set

Detailed Solution for Test: Linear Algebra- 2 - Question 30

There are no solutions.
If we multiply 1st equation by 4, we get
4x + 8y = 20
But 2nd equation says
4x + 8y = 12
Clearly, there can not be any pair of (x,y), which satisfies both equations.

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