GATE AE Linear Algebra: Previous Year Questions

GATE AE (Aerospace Engineering) Linear Algebra previous year questions from the Engineering Mathematics section, with figures and the official answer key where available.

All GATE AE papers

GATE AE 2026, Question 22 (1 mark, MSQ)

Engineering Mathematics - Linear Algebra

An $n \times n$ square matrix $A$ satisfies $A^T = A^{-1}$. The determinant of this matrix may take which of the following value(s)?

  1. $+1$
  2. $-1$
  3. $n$
  4. $0$

Official answer: A, B

GATE AE 2026, Question 32 (1 mark, NAT)

Engineering Mathematics - Linear Algebra

If a matrix can be written as $A = uv^T$, where both $u$ and $v$ are $n$-dimensional real-valued non-zero column vectors, then the rank of the matrix $A$ is ________ (answer in integer).

Official answer: 1

GATE AE 2026, Question 37 (2 marks, MCQ)

Engineering Mathematics - Linear Algebra

For the matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, if the relation $a + b = c + d$ holds and $a, b, c, d \neq 0$, then which one of the following statements about $A$ is FALSE?

  1. $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$ is an eigenvector
  2. $\lambda = a + b$ is an eigenvalue
  3. $\lambda = d - b$ is an eigenvalue
  4. $\lambda = d + b$ is an eigenvalue

Official answer: D

GATE AE 2025, Question 11 (1 mark, MCQ)

Engineering Mathematics - Linear Algebra

For any real symmetric matrix $A$, the transpose of $A$ is _____ .

  1. inverse of $A$
  2. null matrix
  3. $-A$
  4. $A$

Official answer: D

GATE AE 2025, Question 12 (1 mark, MCQ)

Engineering Mathematics - Linear Algebra

The system of equations $$\begin{aligned} 2x + 3y + z &= 0 \\ x + y &= 0 \\ y + z &= 0 \end{aligned}$$ The given system of equations has

  1. a unique solution.
  2. infinitely many solutions.
  3. no solution.
  4. a finite number of solutions.

Official answer: B

GATE AE 2025, Question 13 (1 mark, MCQ)

Engineering Mathematics - Linear Algebra

The eigenvalues of the matrix $\begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}$ are _____.

  1. 0, 2
  2. 2, 3
  3. 1, 3
  4. 1, 2

Official answer: C

GATE AE 2024, Question 11 (1 mark, MCQ)

Engineering Mathematics - Linear Algebra

The following system of linear equations $$\begin{aligned} 7x - 3y + z &= 0 \\ 3x - y + z &= 0 \\ x - y - z &= 0 \end{aligned}$$ has:

  1. infinitely many solutions
  2. a unique solution
  3. no solution
  4. three solutions

Official answer: A

GATE AE 2024, Question 51 (2 marks, NAT)

Engineering Mathematics - Linear Algebra

Consider the matrix $A = \begin{bmatrix} 5 & -4 \\ k & -1 \end{bmatrix}$, where $k$ is a constant. If the determinant of $A$ is 3, then the ratio of the largest eigenvalue of $A$ to the constant $k$ is ___________ (rounded off to 1 decimal place).

Official answer: 1.4 to 1.6

GATE AE 2023, Question 29 (1 mark, MSQ)

Engineering Mathematics - Linear Algebra

Which of the following statement(s) is/are true with respect to eigenvalues and eigenvectors of a matrix?

  1. The sum of the eigenvalues of a matrix equals the sum of the elements of the principal diagonal.
  2. If $\lambda$ is an eigenvalue of a matrix A, then $\dfrac{1}{\lambda}$ is always an eigenvalue of its transpose ($A^T$).
  3. If $\lambda$ is an eigenvalue of an orthogonal matrix A, then $\dfrac{1}{\lambda}$ is also an eigenvalue of A.
  4. If a matrix has $n$ distinct eigenvalues, it also has $n$ independent eigenvectors.

Official answer: A, C, D

GATE AE 2023, Question 31 (1 mark, NAT)

Engineering Mathematics - Linear Algebra

The system of equations $$\begin{aligned} x - 2y + \alpha z &= 0 \\ 2x + y - 4z &= 0 \\ x - y + z &= 0 \end{aligned}$$ has a non-trivial solution for $\alpha =$ ______. (Answer in integer)

Official answer: 3

GATE AE 2023, Question 43 (2 marks, MSQ)

Engineering Mathematics - Linear Algebra

Given the vectors $\vec{A} = 9\hat{i} - 5\hat{j} + 2\hat{k}$, $\vec{B} = 11\hat{i} + 4\hat{j} + \hat{k}$, $\vec{C} = -7\hat{i} + 14\hat{j} - 3\hat{k}$, which of the following statement(s) is/are TRUE?

  1. Vectors $\vec{A}$, $\vec{B}$ and $\vec{C}$ are coplanar
  2. The scalar triple product of the vectors $\vec{A}$, $\vec{B}$ and $\vec{C}$ is zero
  3. $\vec{A}$ and $\vec{B}$ are perpendicular
  4. $\vec{C}$ is parallel to $\vec{A} \times \vec{B}$

Official answer: A, B

GATE AE 2022, Question 12 (1 mark, MCQ)

Engineering Mathematics - Linear Algebra

Let $\hat{i}, \hat{j}$, and $\hat{k}$ be the unit vectors in the x, y and z directions, respectively. If the vector $\hat{i} + \hat{j}$ is rotated about positive $\hat{k}$ by $135^\circ$, one gets

  1. $-\hat{i}$
  2. $-\hat{j}$
  3. $-\dfrac{1}{\sqrt{2}}\hat{j}$
  4. $-\sqrt{2}\,\hat{i}$

Official answer: D

GATE AE 2022, Question 24 (1 mark, MSQ)

Engineering Mathematics - Linear Algebra

If $\hat{a}, \hat{b}, \hat{c}$ are three mutually perpendicular unit vectors, then $\hat{a} \cdot (\hat{b} \times \hat{c})$ can take the value(s)

  1. 0
  2. 1
  3. -1
  4. $\infty$

Official answer: B, C

GATE AE 2022, Question 49 (2 marks, NAT)

Engineering Mathematics - Linear Algebra

The largest eigenvalue of the given matrix is _____. $$\begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{bmatrix}$$

Official answer: 2

GATE AE 2018, Question 11 (1 mark, MCQ)

Engineering Mathematics - Linear Algebra

Let $\vec{a}, \vec{b}$ be two distinct vectors that are not parallel. The vector $\vec{c} = \vec{a} \times \vec{b}$ is

  1. zero.
  2. orthogonal to $\vec{a}$ alone.
  3. orthogonal to $\vec{a} + \vec{b}$.
  4. orthogonal to $\vec{b}$ alone.

Official answer: C

GATE AE 2018, Question 30 (1 mark, NAT)

Engineering Mathematics - Linear Algebra

The determinant of the matrix $\begin{bmatrix} 1 & 1 & -1 \\ 2 & 1 & 0 \\ 3 & 1 & 1 \end{bmatrix}$ is ______ (accurate to one decimal place).

Official answer: 0

GATE AE 2016, Question 19 (1 mark, MCQ)

Engineering Mathematics - Linear Algebra

Consider an eigenvalue problem given by $\mathbf{A}x = \lambda_i x$. If $\lambda_i$ represent the eigenvalues of the non-singular square matrix $\mathbf{A}$, then what will be the eigenvalues of matrix $\mathbf{A}^2$?

  1. $\lambda_i^4$
  2. $\lambda_i^2$
  3. $\lambda_i^{1/2}$
  4. $\lambda_i^{1/4}$

Official answer: B

GATE AE 2016, Question 20 (1 mark, MCQ)

Engineering Mathematics - Linear Algebra

If $\mathbf{A}$ and $\mathbf{B}$ are both non-singular $n \times n$ matrices, then which of the following statement is NOT TRUE. Note: det represents the determinant of a matrix.

  1. $\det(\mathbf{AB}) = \det(\mathbf{A})\det(\mathbf{B})$
  2. $\det(\mathbf{A}+\mathbf{B}) = \det(\mathbf{A}) + \det(\mathbf{B})$
  3. $\det(\mathbf{A}\mathbf{A}^{-1}) = 1$
  4. $\det(\mathbf{A}^T) = \det(\mathbf{A})$

Official answer: B

GATE AE 2016, Question 46 (2 marks, MCQ)

Engineering Mathematics - Linear Algebra

Consider the following system of linear equations: $$\begin{aligned} 2x - y + z &= 1 \\ 3x - 3y + 4z &= 6 \\ x - 2y + 3z &= 4 \end{aligned}$$ This system of linear equations has

  1. no solution.
  2. one solution.
  3. two solutions.
  4. three solutions.

Official answer: A

GATE AE 2014, Question 11 (1 mark, MCQ)

Engineering Mathematics - Linear Algebra

For a real symmetric matrix $[A]$, which of the following statements is true:

  1. The matrix is always diagonalizable and invertible.
  2. The matrix is always invertible but not necessarily diagonalizable.
  3. The matrix is always diagonalizable but not necessarily invertible.
  4. The matrix is always neither diagonalizable nor invertible.

Official answer: C

GATE AE 2014, Question 36 (2 marks, MCQ)

Engineering Mathematics - Linear Algebra

If $[A] = \begin{bmatrix} 3 & -3 \\ -3 & 4 \end{bmatrix}$. Then $\det\left(-[A]^2 + 7[A] - 3[I]\right)$ is

  1. 0
  2. -324
  3. 324
  4. 6

Official answer: A