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   is _____ (rounded off to the nearest integer).     (2024)
(a) 15
(b) 26
(c) 29
(d) 45
Ans: (c)
Sol:

Q2: Which one of the following matrices has an inverse?     (2024)
(a)

(b)
(c) 
(d)
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 =
The matrix A satisfies the equation 6A^-1 = A² + cA + dI where $𝑐$c and d are scalars and I is the identity matrix. Then $(𝑐+𝑑)$(c + d) is equal to    (2022)
(a) 5 
(b) 17
(c) -6
(d) 11
Ans: a
Sol: Characteristic equation:
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: e^A denotes the exponential of a square matrix A. Suppose $𝜆$λ is an eigenvalue and v is the corresponding eigen-vector of matrix A. 
Consider the following two statements:    (2022)
Statement 1: e^λ is an eigenvalue of e^A .
Statement 2: 𝑣 is an eigen-vector of e^A.
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, a_{i, 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 ⇒ a_{ij = (i - i)}³ = 0∀i
for i ≠ j ⇒ a_{ij = (i - j)}³  = (-(j - i))³ = - (j - i)³ = -a_ji
∴ A_3×3 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 I_A = 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 p² + q² = 1. The eigenvalues of the matrix  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
∣A_2×2 − 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)
(a) 5
(b) 6
(c) 8
(d) 9
Ans: d
Sol: As per GATE official answer key MTA (Marks to ALL)
consider, u_{11 =} u₂₂ = u_{33 = 1}
The number of purely real elements of lower triangular matrix are 9.

Q9: Consider a 2 x 2 matrix M = [v₁ v₂], where, v₁ and v₂ are the column vectors. Suppose  where  are the row vectors. Consider the following statements:
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: 
Statement (1) and (2) are both correct. Option (C) is correct.

Q10: The rank of the matrix, is ______      (2019)
(a) 1
(b) 2
(c) 3
(d) 4
Ans: (c)
Sol:
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  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:
(1 − λ)((2 − λ)(−2 − λ))−1(0 − 0) = 0
λ = 1, 2, −2

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

Q14: The eigen values of the matrix given below are       (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.
λ= -1, -3
λ = (0, -1, -3)

Q15: The matrix  has three distinct eigenvalues and one of its eigenvectors is Which one of the following can be another eigenvector of A?     (SET-1 (2017))
(a)
(b)
(c)
(d)
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 The corresponding orthogonal vector in the given option is C i.e.
Q16: Let  Consider the set S of all vectors such that a² + b² = 1 where  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:
⇒
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 A^TA is less than 2.
(b) Rank of A^TA is equal to 2.
(c) Rank of A^TA is greater than 2.
(d) Rank of A^TA can be any number between 1 and 3.
Ans: (b)
Sol: Result,
Rank(A^TA) = Rank(A)

Q19:  Let the eigenvalues of a 2 x 2 matrix A be 1, -2 with eigenvectors x₁ and x₂ respectively. 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; x₁ + x_2, x_{1 -} x₂
(c) 2, 0; x₁, x₂
(d) 2, 0; x₁ + x_2, x_{1 -} x₂
Ans: (a)
Sol: Eigen value of A² − 3A + 4I are
(1)² − 3(1) + 4 and (−2)² − 3(−2) + 4
= 2, 14
Note: A²X = l²X  
⇒ X is eigen vector for A² corresponding to eigen value l²
X₁ and  X₂ are eigen vector of A corresponding to 1, -2
Then X₁ and X₂ are eigen vector of 𝐴² − 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:
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  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)
(c)
(d)
Ans: (a)
Sol:
If  then all the row (column) vectors of A are linearly independent.

Q22: The maximum value of "a" such that the matrix has three linearly independent real eigenvectors is    (SET-1(2015))
(a)
(b)
(c)
(d)
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
Given a + d = −6
∣A∣ = ad − b² 
Now since b² is always non-negative, maximum determinant will come when $𝑏2=0$b² = 0. 
So we need to maximize
⇒ a = -3 is the only stationary point
Since,  we have a maximum at $𝑎=-3$a = −3. 
Since  a + d = −6, corresponding value of 𝑑 = −3.
 ∣A∣ = ad = −3 × −3 = 9.

Q24: Two matrices A and B are given below:
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:
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)