The GATE Grind

GATE 2026 EE – Question 55

Engineering Mathematics · Linear Algebra: Matrix Algebra · 2 marks · Multiple select

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?

  1. The value of the determinant of $A$ is either +1 or −1
  2. The eigenvalues of $A$ have modulus 1
  3. $\|A\boldsymbol x\|=\|\boldsymbol x\|$, for all $\boldsymbol x\in R^n$, where $\|\boldsymbol x\|$ denotes the Euclidean norm of $\boldsymbol x$, and $(A\boldsymbol x)^T(A\boldsymbol y)\ne\boldsymbol x^T\boldsymbol y$, for all distinct $\boldsymbol x,\boldsymbol y\in R^n$
  4. $\|A\boldsymbol x\|=\|\boldsymbol x\|$, for all $\boldsymbol x\in R^n$, where $\|\boldsymbol x\|$ denotes the Euclidean norm of $\boldsymbol x$, and $(A\boldsymbol x)^T(A\boldsymbol y)=\boldsymbol x^T\boldsymbol y$, for all $\boldsymbol x,\boldsymbol y\in R^n$

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Correct answer: (A) The value of the determinant of $A$ is either +1 or −1; (B) The eigenvalues of $A$ have modulus 1; (D) $\|A\boldsymbol x\|=\|\boldsymbol x\|$, for all $\boldsymbol x\in R^n$, where $\|\boldsymbol x\|$ denotes the Euclidean norm of $\boldsymbol x$, and $(A\boldsymbol x)^T(A\boldsymbol y)=\boldsymbol x^T\boldsymbol y$, for all $\boldsymbol x,\boldsymbol y\in R^n$

Explanation

An orthogonal matrix satisfies $A^TA=I$, so $\det(A)^2=1$ and $\det A=\pm1$ (A). It preserves lengths, so any eigenvalue satisfies $|\lambda|\,\|\boldsymbol v\|=\|A\boldsymbol v\|=\|\boldsymbol v\|$, hence $|\lambda|=1$ (B). Inner products are preserved: $(A\boldsymbol x)^T(A\boldsymbol y)=\boldsymbol x^TA^TA\boldsymbol y=\boldsymbol x^T\boldsymbol y$ (D), which makes C false.