Linear Algebra - 1


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10 Questions MCQ Test Question Bank for GATE Computer Science Engineering | Linear Algebra - 1

Linear Algebra - 1 for Computer Science Engineering (CSE) 2022 is part of Question Bank for GATE Computer Science Engineering preparation. The Linear Algebra - 1 questions and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus.The Linear Algebra - 1 MCQs are made for Computer Science Engineering (CSE) 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Linear Algebra - 1 below.
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Linear Algebra - 1 - Question 1

 A square matrix is singular whenever:

Detailed Solution for Linear Algebra - 1 - Question 1

If the rows (or columns) of a square matrix are linearly dependent, then the determinant of matrix becomes zero.
Therefore, whenever the rows are linearly dependent, the matrix is singular.

Linear Algebra - 1 - Question 2

If A and B are real symmetric matrices of size n x n. Then, which one of the following is true?

Detailed Solution for Linear Algebra - 1 - Question 2

The matrix M is s aid to be sym m etric iff MT= M

Linear Algebra - 1 - Question 3

Let a = (aij) be an n-rowed square matrix and I12 be the matrix obtained by interchanging the first and second rows of the n-rowed Identity matrix. Then AI12 is such that its first

Detailed Solution for Linear Algebra - 1 - Question 3

Let, 
I12 is the matrix obtained by inter-changing the first and second row of the identity Matrix I.
So

AI12 is the matrix having first column same as the second column of A.

Linear Algebra - 1 - Question 4

The rank of the matrix given below is:

Detailed Solution for Linear Algebra - 1 - Question 4

The given matrix is:

Linear Algebra - 1 - Question 5

Consider the following determinant:

Which of the following is a factor of Δ?

Detailed Solution for Linear Algebra - 1 - Question 5

The determinant of a matrix can’t be affected by elementary row operation
So,



So (a - b) is a factor of Δ.

Linear Algebra - 1 - Question 6

 The rank of the matrix  is

Detailed Solution for Linear Algebra - 1 - Question 6

The given matrix is 
Above matrix has only 1 independent row, so the given matrix has rank 1.

Linear Algebra - 1 - Question 7

The following system of equations:

Has a unique solution. The only possible value(s) for a is/are

Detailed Solution for Linear Algebra - 1 - Question 7

The augmented matrix for above system is


Now as long as a - 5 ≠ 0, 
rank (A) = rank (A | B) = 3 
∴ a can take any real value except 5.

Linear Algebra - 1 - Question 8

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 = O has a nontrivial solution.
S4: M has an inverse.

Detailed Solution for Linear Algebra - 1 - Question 8

S1 and S2:
Since M has zero determinant, its rank is not full i.e. if M is of size 3 x 3 , then its rank is not 3. So there is a linear combination of rows which evaluates to 0 i.e. k1R1 + k2R+...+knRn = 0 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.
So S1 and S2 are true.

Linear Algebra - 1 - Question 9

Consider the matrix as given below:

Which one of the following options provides the CORRECT values of the eigenvalues of the matrix?

Detailed Solution for Linear Algebra - 1 - Question 9

Since the given matrix is upper triangular, its eigen values are the diagonal elements themselves, which are 1, 4 and 3.

Linear Algebra - 1 - Question 10

Consider the following 2 x 2 matrix A where two elements are unknown and are marked by ‘a’ and ‘b'. The eigenvalues of this matrix are -1 and 7. What are the values of ‘a’ and 'b'?

Detailed Solution for Linear Algebra - 1 - Question 10

Trace = Sum of eigen values
1 + a = 6
⇒ a = 5
Determinant = Product of eigen values 

⇒ b = 3
∴ a = 5, b = 3

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