Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE) PDF Download

Q1: Consider two matrices
The determinant of the matrix AB is ______ (in integer).   [2024, Set-ll]
Ans: 10 to 10
Q2: The statements P and Q are related to matrices A and B, which are conformable for both addition and multiplication. 
P : (A + B)^⊤ = A^⊤ + B^⊤
Q : (AB)^⊤ = A^⊤ B^⊤  
Which one of the following options is CORRECT? [2024, Set-ll]
(a) Both P and Q are FALSE 
(b) Both P and Q are TRUE 
(c) P is FALSE and Q is TRUE 
(d) P is TRUE and Q is FALSE
Ans: (d)
According to properties of a matrix 
(i) (A+B)^⊤ = A^⊤ + B^⊤
The sum of transpose of matrices is equal to the transpose of the sum of two matrices. 
(ii) (AB)^⊤ = B^⊤ A^⊤
The product of the transpose of two matrices in reverse order is equal to the transpose of the product of them.

Q3: What are the eigenvalues of the matrix [2024, Set-l]
(a) -5,-1,2
(b)-5,1,2
(c) 1,3,4
(d) 1,2,5
Ans: (d)
λ₁ + λ₂ + λ₃ = 8
λ₁ λ₂ λ₃ = 8 = ∣A∣ = 10 
Only option (A) satisfied above condition. 
1 + 2 + 5 = 8 
1 × 2 × 5 = 10

Q4: Cholesky decomposition is carried out on the following square matrix [A].

Let I_ij and a_ij be the (i, j)^th elements of matrices [L] and [A], respectively. If the element I₂₂  of the decomposed lower triangular matrix [L] is 1.968 , what is the value (rounded off to the nearest integer) of the element a₂₂ ?  [2023, Set-ll]
(a) 5
(b) 7
(c) 9
(d) 11
Ans: (b)
We know, cholesky decomposition, A = LL^T
Where, L= lower tringular matrix

On comparison on both sides,

Q5: For the matrix

which of the following statements is/are TRUE? [2023, Set-ll]
(a) [A] {X} = {b} has a unique solution 
(b) [A] {X} = {b} does not have a unique solution 
(c) [A] has three linearly independent eigenvectors 
(d) [A] is a positive definite matrix
Ans: (b,c)

So AX = B does not have unique solution because ρ(A) < 3

Matrix A has three distinct Eigen values so have three linearly independent eigen vectors. so option (C) is correct. Given matrix is symmetric matrix with real value entries. Hence A is not a positive definite matrix. because

Hence option (D) is incorrect.

Q6: For the matrix

Which of the following statements is/are TRUE? [2023, Set-l]
(a) The eigenvalues of [A] ^T are same as the eigenvalues of [A] 
(b) The eigenvalues of [A]⁻¹ are the reciprocals of the eigenvalues of [A] 
(c) The eigenvectors of [A]^T are same as the eigenvectors of [A] 
(d) The eigenvectors of [A]⁻¹ are same as the eigenvectors of [A]
Ans: (a, b, c, d)
Ax = λx… (i) 
A^Tx = λx… (ii) 
A and A^T both have same eigen values and eigen vectors. 
Ax = λx…(i) 
⇒ A⁻¹ Ax = A⁻¹ (λx) = λA⁻¹ x 
⇒ x = λ A⁻¹ x 

So, eigen value and eigen vector of and x.

Q7: If M is an arbitrary real n × n matrix, then which of the following matrices will have non-negative eigenvalues?  [2024, Set-l]
(a) M²
(b) MM^T
(c) M^T M 
(d) (MT)²
Ans: (a, b, c, d)





[λ² is eigen value of (M^⊤)² which non negative] Hence, option A, B, C, D are correct.

Q8: Let y be a non-zero vector of size 2022 x 1. Which of the following statement(s) is/are TRUE?   [2022, Set-ll]
(a) yy^T is a symmetric matrix. 
(b) y^Ty is an eigenvalue of yy^T
(c) yy^T has a rank of 2022. 
(d) yy^T is invertible.
Ans: (a,b)
Let vector


From above information 
yy^T is asymmetric. 
y^Ty is an eigen value of yy^T. 
yy^T has rank 1 
det(yy^T) = 0 so, yy^T is not invertible.

Q9: P and Q are two square matrices of the same order. Which of the following statement(s) is/are correct?  [2022, Set-ll]
(a) If P and Q are invertible, then [PQ]⁻¹ =Q⁻¹ P⁻¹
(b) If P and Q are invertible, then [QP]⁻¹ =P ⁻¹ Q⁻¹
(c)If P and Q are invertible, then [PQ]⁻¹ = Q⁻¹ P⁻¹
(d) If P and Q are not invertible, then [PQ]⁻¹ = P⁻¹ Q⁻¹  
Ans: (a, b)
If P and Q are invertible then (PQ)⁻¹ = Q⁻¹ P⁻¹ is correct. Let,

Hence, proved. Similarly, we can prove if P, Q are invertible then (QP)⁻¹ = P⁻¹ Q ⁻¹ 

Q10: The matrix M is defined asand has eigenvalues 5 and -2. The matrix Q is formed as Q = M³ − 4M² − 2M 
Which of the following is/are the eigenvalue(s) of matrix Q ?  [2022, Set-l]
(a)15 
(b) 25 
(c) -20 
(d) -30
Ans: (a, c)
Eigen values of M are 5, -2. So, eigen value of Q = M³ − 4M² −2M are 
5³ − 4 × 5² − 2 × 5 = 15 
(−2)³ − 4 × (−2)² − 2 × (−2) =−20 

Q11: The smallest eigenvalue and the corresponding eigenvector of the matrix , respectively, are  [2021, Set-ll]
(a) 1.55 and
(b) 2.00 and
(c) 1.55 and
(d) 1.55 and
Ans: (a)




Q12: If A is a square matrix then orthogonality property mandates  [2021, Set-ll]
(a) AA^T= I
(b) AA^T = 0
(c) AA^T= A⁻¹
(d) AA^T = A²
Ans: (a)
If, AA^⊤ = I or A⁻¹ = A^T
The matrix is orthogonal.

Q13: The rank of the matrix [2021, Set-ll]
(a) 1
(b) 2
(c) 3
(d) 4
Ans: (c)

Rank(A) = 3

Q14: If P = and Q =  then QTPT is [2021, Set-l]
(a)
(b)
(c)
(d)
Ans: (d)

Now using Reversal law


Q15: The rank of matrix [2021, Set-l]
(a) 1
(b) 2
(c) 3
(d) 4
Ans: (b)


Q16: A 4x4 matrix [P] is given below

The eigen values of [P] are  [2020, Set-ll]
(a) 0,3,6,6
(b) 1,2,3,4
(c) 3,4,5,7
(d) 1,2,5,7
Ans: (d)
|P|= 70 and Trace (P) = 15
So, only option, (1, 2, 5, 7) satisfies.

Q17: Consider the system of equations

The value of x₃ (round off to the nearest integer), is _______. [2020, Set-l]
(a) 1 
(b) 2 
(c) 3 
(d) 4
Ans: (c)


Q18: Consider the hemispherical tank of radius 13 m as shown in the figure (not drawn to scale). What is the volume of water (in m³) when the depth of water at the centre of the tank is 6 m?   [2019: 2 Marks, Set-ll]

(a) 156π
(b) 396π
(c) 468π
(d) 78π
Ans: (b)
Volume of water

Q19: The inverse of the matrix [2019: 2 Marks, Set-ll]

Ans: (d)

Q20: The rank of the following matrix is  [2018 : 2 Marks, Set-II]

(a) 1
(b) 2
(c) 3
(d) 4
Ans: (b)

Number of non zero rows = 2
Rank of A = 2

Q21: The matrix has  [2018 : 2 Marks, Set-II]
(a) real eigenvalues and eigenvectors
(b) real eigenvalues but complex eigenvectors
(c) complex eigenvalues but real eigenvectors
(d) complex eigenvalues and eigenvectors
Ans: (d)

∴Complex Eigenvalues and complex Eigen vectors.

Q22: Which one of the following matrices is singular?  [2018: 1 Marks, Set-I]




Ans: (c)
Option (a): |A| = 6 - 5 = 1
Option (b): |A| = 9 - 4 = 5
Option (c): |A| = 12-12 = 0
Option (d): |A| = 8 - 18 = -10
Hence matrix (c) is singular.

Question for Linear Algebra

Try yourself:For the given orthogonal matrix Q,    [2018: 1 Marks, Set-I]

The inverse is

• A.

  
• B.

  
• C.

  
• D.

  

Explanation

Or ∵ Q is orthogonal.
∴ Q^-1 = Q^T 

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Q23: If  A = and B = AB^T is equal to  [2017 : 2 Marks, Set-II]

Ans: (a)


Q24: Consider the following simultaneous equations (with c₁ and c₂ being constants):  [2017 : 1 Mark, Set-II]
3x₁ + 2x₂ = c₁
4x₁ + x₂ = c₂
The characteristics equation for these simultaneous equations is
(a) λ² - 4λ - 5 = 0
(b) λ² - 4λ + 5 = 0
(c) λ² + 4λ - 5 = 0
(d) λ² + 4λ + 5 = 0
Ans: (a)


|A - λI| = 0
(3 - λ) (1 - λ) - 8 = 0
3 - 4λ + λ² - 8 = 0
λ² - 4λ - 5 = 0

Q25: The matrix P is the inverse of a matrix Q. If I denotes the identity matrix, which one of the following options is correct?  [2017: 1 Mark, Set-I]
(a) PQ = I but QP ≠ I
(b) QP = I but PQ ≠ I
(c) PQ = I and QP = I
(d) PQ = QP = I
Ans:(c)
Given that P is inverse of Q.
P = Q^-1  
PQ = Q^-1Q,  QP = QQ^-1
PQ = I , QP = I
∴ PQ = QP = I

Question for Linear Algebra

Try yourself:The two Eigenvalues of the matrix  have a ratio of 3 : 1 for p = 2. What is another value of p for which the Eigenvalues have the same ratio of 3 : 1?   
[2015: 2 Marks, Set-II]

• A.

  -2
• B.

  1
• C.

  7 / 3
• D.

  14 / 3

Explanation

Let λ₁ and λ₂ be the eigenvalue of matrix A
∴ 
Sum of eigen value:
= λ₁ + λ₂ = 2 + p     .......(i)
Product of eigen value:
λ₁λ₂ = 2p - 1   ........(ii)






By putting values of p from options.
By putting option (d) 14/3 in above equations gives value.
Hence the ratio of two eigenvalues 

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Question for Linear Algebra

Try yourself:Let A = with n > 3 and a_ij = i.j. The rank of A is    [2015 : 1 Mark, Set-II]

• A.

  0
• B.

  1
• C.

  n - 1
• D.

  n

Explanation

Rank of A = 1
Because each row will be scalar multiple of first row. So we will get only one non-zero row in row Echeleaon form of A.
Alternative:
Rank of A = 1
Because all the minors of order greater than 1 will be zero.

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Question for Linear Algebra

Try yourself:The smallest and largest Eigen values of the following matrix are     [2015 : 2 Marks, Set-I]

• A.

  1.5 and 2.5
• B.

  0.5 and 2.5
• C.

  1.0 and 3.0
• D.

  1.0 and 2.0

Explanation

For eigen values |A - λI| = 0

Only 1 and 2 satisfy this equation.
λ = 1, 1, 2
Hence,Smallest eigen value = 1 and
Largest eigen value = 2

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Q26: For what value of p the following set of equations will have no solution?    [2015 : 1 Mark, Set-I]
2x + 3y = 5
3x + py = 10
Ans: Given system of equations has no solution if the lines are parallel i.e., their slopes are equal
2/3 = 3/p
⇒ p = 4.5

Q27: The rank of the matrix is _____.    [2014 : 2 Marks, Set-II]
Ans:


Determinant of matrix  is not zero.
∴ Rank is 2

Q28: The determinant of matrix [2014 : 1 Mark, Set-II]
Ans:







Interchanging column 1 and column 2 and taking transpose,

Q29: With reference to the conventional Cartesian (x, y) coordinate system, the vertices of a triangle have the following coordinates; (x₁, y₁) = (1, 0); (x₂, y₂) = (2, 2); (x₃, y₃) = (4, 3). The area of the triangle is equal to    [2014 : 1 Mark, Set-I]
(a) 3/2
(b) 3/4
(c) 4/5
(d) 5/2
Ans: (a)

Area of triangle is

Q30: The sum of Eigen values of matrix, [M] is where [2014 : 1 Mark, Set-I]
(a) 915
(b) 1355
(c) 1640
(d) 2180
Ans: (a)
Sum of eigen values = trace of matrix
= 215 + 150 + 550 = 915

Q31: Given the matrices the product K^T JK is ____.    [2014 : 1 Mark, Set-I]
Ans:

Q32: There are three matrixes P(4 x 2), Q(2 x 4) and R(4 x 1). The minimum of multiplication required to compute the matrix PQR is    [2013 : 1 Mark]
Ans:If we multiply QR first then,
Q1:_2x4 x R_(4x1) having multiplication number 8.
There fore P_{(4 x 2)} QR_{(2 x 1)} will have minimum number of multiplication = (8 + 8) = 16.

Q33: The eigen values of matrix [2011 : 2 Marks]
(a) -2.42 and 6.86
(b) 3.48 and 13.53
(c) 4.70 and 6.86
(d) 6.86 and 9.50
Ans: (b)
We need eigen values of

The characteristic equation is,

So eigen values are,

λ = 3.48, 13.53

Q34: [A] is square matix which is neither symmetric nor skew-symmetric and [A]^T is its transpose. The sum and difference of these matrices are defined as [S] = [A] + [A]^T and [D] = [A] - [A]^T, respectively. Which of the following statements is TRUE?    [2011 : 1 Mark]

(a) Both [S] and [D] are symmetric
(b) Both [S] and [D] are skew-symmetric
(c) [S] is skew-symmetric and [D] is symmetric
(d) [S] is symmetric and [D] is skew-symmetric
Ans: (d)
Since (A + A^t) = A^t + (A^t)^t
= A^t + A
i.e. S^t = S
∴ S is symmetric
Since (A - A^t)^t = A^t - (A^t)^t
= A^t - A = -(A - A^t)
i.e. D^t = - D
So D is Skew-Symmetric.

Q35: The inverse of the matrix [2010 : 2 Marks]

Ans: (b)