Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) PDF Download

Q1: The sum of the eigenvalues of the matrix  Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) is _____ (rounded off to the nearest integer).     (2024)
(a) 15
(b) 26
(c) 29
(d) 45
Ans:
(c)
Sol:
Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q2: Which one of the following matrices has an inverse?     (2024)
(a) Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

(b) Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(c) Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(d) Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Ans: (c)
Sol:
Option :A
det = 0
Cannot have inverse.
Option : B
det = 0
Cannot have inverse.
Option : D
det = 0
It cannot have inverse.
Option : C
∣𝐴∣ ≠ 0
∴ It has inverse.

Q3: Consider a matrix A = Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
The matrix A satisfies the equation 6A-1 = A+ cA + dI where 𝑐c and d are scalars and I is the identity matrix. Then (𝑐+𝑑)(d) is equal to    (2022)
(a) 5 
(b) 17
(c) -6

(d) 11
Ans: a
Sol: Characteristic equation:
Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)By cayley hamilton theorem
A3 - 6A2 + 11A - 6 = 0
A2  - 6A + 11I = 6A-1
On comparison : c = -6 and d = 11
Therefore, c + d = -6 + 11 = 5

Q4: eA denotes the exponential of a square matrix A. Suppose 𝜆λ is an eigenvalue and v is the corresponding eigen-vector of matrix A. 
Consider the following two statements:    (2022)
Statement 1: eλ 
is an eigenvalue of eA .
Statement 2: 𝑣 is an eigen-vector of 
eA.
Which one of the following options is correct?

(a) Statement 1 is true and statement 2 is false.
(b) Statement 1 is false and statement 2 is true
(c) Both the statements are correct.
(d) Both the statements are false.
Ans: c

Sol: Eigen value will change but eigen vector not change.

Q5: Consider a 3 x 3 matrix A whose (i, j)-th element, ai, j = (i - j)3. Then the matrix A will be    (2022)
(a) symmetric.
(b) skew-symmetric.
(c) unitary
(d) null.
Ans:
(b)
Sol:
for i = j ⇒ aij = (i - i)= 0∀i
for i ≠ j ⇒ aij = (i - j) = (-(j - i))3 = - (j - i)3 = -aji
∴ A3×3 is skew symmetric matrix.

Q6: Let A be a 10 × 10 matrix such that  A5 is a null matrix, and let I be the 10 × 10 identity matrix. The determinant of A + I is _______.     (2021)
(a) 1
(b) 2
(c) 4
(d) 8
Ans: 
(a)
Sol: Given: A5 = 0
Ax = λx
⇒ A5x = λ5x (∵ x = 0)
⇒ λ5 = 0
⇒ λ = 0
Eigen values of A + I  given λ + 1
∵ Eigen values of IA = 1
Hence |A + I| = Product of eigen values = 1 x 1 x 1 x .... 10 times
= 1

Q7: Let p and q be real numbers such that p+ q2 = 1. The eigenvalues of the matrix Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) are    (2021)
(a) 1 and 1
(b) 1 and -1
(c) j and -j
(d) pq and -pq
Ans:
(b)
Sol: Characteristic equation of A
∣A2×2 − λI∣ = (−1)2 λ2 + (−1)1 Tr(A)λ + ∣A∣ = 0
λ2 −(p − p)λ + (−p2 − q2) = 0
⇒ λ2 - 1 = 0
⇒ λ = ± 1

Q8: The number of purely real elements in a lower triangular representation of the given 3 x 3 matrix, obtained through the given decomposition is    (2020)
Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)(a) 5
(b) 6
(c) 8
(d) 9
Ans:
d
Sol: As per GATE official answer key MTA (Marks to ALL)
Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)consider, u11 = u22 = u33 = 1
Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)The number of purely real elements of lower triangular matrix are 9.

Q9: Consider a 2 x 2 matrix M = [v1 v2], where, vand vare the column vectors. Suppose Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) where Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) are the row vectors. Consider the following statements:
Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Which of thefollowing options is correct?    (2019)
(a) Statement 1 is true and statement 2 is false
(b) Statement 2 is true and statement 1 is false
(c) Both the statements are true
(d) Both the statements are false
Ans:
(c)
Sol: 
Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Statement (1) and (2) are both correct. Option (C) is correct.

Q10: The rank of the matrix, Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)is ______      (2019)
(a) 1
(b) 2
(c) 3
(d) 4
Ans: 
(c)
Sol:
Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Which is in echelon form
∴ ρ(A) = 3.

Q11: M is a 2 x 2 matrix with eigenvalues 4 and 9. The eigenvalues of M2 are    (2019)
(a) 4 and 9
(b) 2 and 3
(c) -2 and -3
(d) 16 and 81
Ans:
(d)
Sol: M is 2 x 2 matrix with eigen values 4 and 9. The eigen values of M2 are 16 and 81.

Q12: Let Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) and B = A3 - A2 - 4A + 5I, where I is the 3 x 3 identity matrix. The determinant of B is _____ (up to 1 decimal place).     (2018)
(a) 0.5
(b) 1.0
(c) 2.8
(d) 2.1
Ans:
(b)
Sol:
Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)(1 − λ)((2 − λ)(−2 − λ))−1(0 − 0) = 0
λ = 1, 2, −2
Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q13: Consider a non-singular 2 x 2 square matrix A. If trace(A) = 4 and trace (𝐴2) = 5, the determinant of the matrix A is ______(up to 1 decimal place).    (2018)
(a) 2.5
(b) 3.5
(c) 1.2
(d) 5.5
Ans: 
d
Sol: A is 2 x 2 matrix
Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q14: The eigen values of the matrix given below are     Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)  (SET-2 (2017))
(a) (0, -1, -3)
(b) (0, -2, -3)
(c) (0, 2, 3)
(d) (0, 1, 3)
Ans:
(a)
Sol: The characteristics equation is |A - λ|i = 0.
Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)λ= -1, -3
λ = (0, -1, -3)

Q15: The matrix Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) has three distinct eigenvalues and one of its eigenvectors is Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Which one of the following can be another eigenvector of A?     (SET-1 (2017))
(a) Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

(b)Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(c)Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(d)Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Ans: (c)
Sol: The given matrix is symmetric and all its eigen values are distinct. Hence all its eigen vectors are orthogonal one of the eigen vector is Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)The corresponding orthogonal vector in the given option is C i.e. Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q16: Let Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) Consider the set S of all vectors Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)such that a2 + b2 = 1 where Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) Then S is    (SET-2 (2016))
(a) a circle of radius √10
(b) a circle of radius 1/√10
(c) an ellipse with major axis along (1/1)
(d) an ellipse with minor axis along (1/1)
Ans:
(d)
Sol:
Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Ellipse with major axis along [1/1].

Q17: A 3 x 3 matrix P is such that,  P3 = P. Then the eigenvalues of P are    (SET-2(2016))
(a) 1, 1, -1
(b) 1, 0.5 + j0.866, 0.5 - j0.866
(c) 1, -0.5 + j0.866, -0.5 - j0.866
(d) 0, 1, -1
Ans:
(a)
Sol: By Calyey Hamilton theorem,
λ3 = λ
λ = 0, 1, −1

Q18: Let A be a 4 x 3 real matrix with rank 2. Which one of the following statement is TRUE?    (SET-1(2016))
(a) Rank of ATA is less than 2.
(b) Rank of ATA is equal to 2.
(c) Rank of ATA is greater than 2.
(d) Rank of ATA can be any number between 1 and 3.
Ans:
(b)
Sol: Result,
Rank(ATA) = Rank(A)

Q19:  Let the eigenvalues of a 2 x 2 matrix A be 1, -2 with eigenvectors x1 and xrespectively. Then the eigenvalues and eigenvectors of the matrix A2 - 3A + 4l would, respectively, be     (SET-1 (2016))
(a) 2, 14; x1, x2
(b) 2, 14; x+ x2, x1 - x2
(c) 2, 0; x1, x2
(d) 2, 0; x+ x2, x1 - x2
Ans:
(a)
Sol: Eigen value of A− 3A + 4I are
(1)− 3(1) + 4 and (−2)− 3(−2) + 4
= 2, 14
Note: A2X = l2X  
⇒ X is eigen vector for A2 corresponding to eigen value l2
X1 and  Xare eigen vector of A corresponding to 1, -2
Then X1 and X2 are eigen vector of 𝐴− 3 + 4𝐼 corresponding to 2, 14.  

Q20: Consider a 3 x 3 matrix with every element being equal to 1. Its only non-zero eigenvalue is ____.    (SET-1(2016))
(a) 1
(b) 2
(c) 3
(d) 4
Ans:
(c)
Sol:
Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Eigen value are 0,0,3.



Q21: We have a set of 3 linear equations in 3 unknowns. 'X ≡ Y' means X and Y are equivalent statements and  Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)means X and Y are not equivalent statements.
P : There is a unique solution.
Q : The equations are linearly independent.
R : All eigenvalues of the coefficient matrix are nonzero.
S : The determinant of the coefficient matrix is nonzero.
Which one of the following is TRUE?    (SET-2(2015))
(a) P ≡ Q ≡ R ≡ S
(b) Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

(c) Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(d) Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Ans: (a)
Sol:
Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)If Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) then all the row (column) vectors of A are linearly independent.

Q22: The maximum value of "a" such that the matrix Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)has three linearly independent real eigenvectors is    (SET-1(2015))
(a)Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

(b)Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(c)Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(d)Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Ans: (b)

Q23: If the sum of the diagonal elements of a 2 x 2 matrix is -6, then the maximum possible value of determinant of the matrix is ________.      (SET-1(2015))
(a) 6
(b) 8
(c) 9
(d) 12
Ans:
(c)
Sol: Consider a symmetric matrix Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Given a + d = −6
∣A∣ = ad − b2 
Now since b2 is always non-negative, maximum determinant will come when 𝑏2=0b0. 
So we need to maximize
Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)⇒ a = -3 is the only stationary point
Since, Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) we have a maximum at 𝑎=3a = −3. 
Since  a + d = −6, corresponding value of 𝑑 = −3.
 ∣A∣ = ad = −3 × −3 = 9.

Q24: Two matrices A and B are given below:
Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)If the rank of matrix A is N, then the rank of matrix B is    (SET-3 (2014))
(a) N/2
(b) N-1
(c) N
(d) 2N
Ans:
(c)
Sol:
Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Rank of amtrix does not change when we squaring the matrix, hence rank of B = rank of A = N.

Q25: Which one of the following statements is true for all real symmetric matrices?     (SET-2 (2014))
(a) All the eigenvalues are real.
(b) All the eigenvalues are positive.
(c) All the eigenvalues are distinct.
(d) Sum of all the eigenvalues is zero.
Ans:
(a)

The document Previous Year Questions- Linear Algebra - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Engineering Mathematics for Electrical Engineering.
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FAQs on Previous Year Questions- Linear Algebra - 1 - Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

1. What is the definition of a matrix in linear algebra?
Ans. A matrix in linear algebra is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It is often used to represent systems of linear equations and transformations.
2. How do you determine the rank of a matrix?
Ans. The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. It can be determined by performing row operations to reduce the matrix to row-echelon form and counting the number of non-zero rows.
3. What is the significance of eigenvalues and eigenvectors in linear algebra?
Ans. Eigenvalues and eigenvectors are crucial concepts in linear algebra as they help in understanding the behavior of linear transformations. Eigenvalues represent scalar factors by which eigenvectors are scaled during a transformation, providing insights into the transformation's behavior.
4. How are determinants used in linear algebra?
Ans. Determinants are used in linear algebra to determine invertibility, solve systems of linear equations, calculate areas/volumes, and understand transformations. The determinant of a matrix is a scalar value that provides information about the matrix's properties.
5. Can you explain the concept of linear independence in the context of vectors?
Ans. In linear algebra, a set of vectors is said to be linearly independent if no vector in the set can be represented as a linear combination of the others. Linear independence is essential in determining the dimension of vector spaces and solving systems of linear equations.
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