GATE EE 2026 | Linear Algebra Question 4 | Engineering Mathematics

GATE EE 2026

MCQ (More than One Correct Answer)

-0

Consider an $n \times n$ orthogonal matrix $A$ with real entries and each column having unit Euclidean norm. Which of the following statements is/are correct?

The value of the determinant of $A$ is either +1 or -1 .

The eigenvalues of $A$ have modulus 1.

$\|A x\|=\|x\|$, for all $x \in R^n$, where $\|x\|$ denotes the Euclidean norm of $x$, and $(A x)^{\top}(A y) \neq x^{\top} y$, for all distinct $x, y \in R^n$.

$\|A x\|=\|x\|$, for all $x \in R^n$, where $\|x\|$ denotes the Euclidean norm of $x$, and $(A x)^T(A y)=x^T y$, for all distinct $x, y \in R^n$.

GATE EE 2026

MCQ (More than One Correct Answer)

-0

Consider the system of linear equations: $A x=b$, where $A$ is an $\mathrm{n} \times \mathrm{n}$ matrix, and $x$ and $b$ are $n$-dimensional column vectors.

Suppose this system of equations has a unique solution. Which of the following statements is/are correct?

$A^{-1}$ exists

The system of equations $A^m x=b$ also has a unique solution for $m=1,2,3, \ldots$

$\operatorname{rank}(\mathrm{A})=\operatorname{rank}\left(\mathrm{A}^{\mathrm{m}}\right)$, for $m=1,2,3, \ldots$

$\operatorname{rank}(A)<\operatorname{rank}([A \mid b])$, where $[A \mid b]$ denotes the augmented matrix.

GATE EE 2022

MCQ (Single Correct Answer)

-0.67

e⁴ denotes the exponential of a square matrix A. Suppose $$\lambda$$ is an eigen value and v is the corresponding eigen-vector of matrix A.

Consider the following two statements:

Statement 1 : e^$$\lambda$$ is an eigen value of e^A.

Statement 2 : v is an eigen-vector of e^A.

Which one of the following options is correct?

Statement 1 is true and statement 2 is false.

Statement 1 is false and statement 2 is true.

Both the statements are correct.

Both the statements are false.

GATE EE 2022

MCQ (Single Correct Answer)

-0.67

Consider a matrix $$A = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 4 & { - 2} \cr 0 & 1 & 1 \cr } } \right]$$. The matrix A satisfies the equation 6A^$$-$$1 = A² + cA + dI, where c and d are scalars and I is the identify matrix. Then (c + d) is equal to

$$-$$6

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