Question Bank • QB_PARAMETRIC_EM_MAT_114MSQ • 2 Marks
Section 1: Engineering Mathematics
Linear Algebra: Matrices & Determinants
Subtopic: Orthogonal matrices
Question Statement:
Let $Q$ be an $n \times n$ real orthogonal matrix ($Q^T Q = I$). Which of the following statements is/are ALWAYS TRUE? (Variation QB_PARAMETRIC_EM_MAT_114)
Options:
(A) The determinant of $Q$ can only be $+1$ or $-1$
(B) The inverse of $Q$ is equal to its transpose ($Q^{-1} = Q^T$)
(C) The Euclidean length of any vector $x \in \mathbb{R}^n$ is preserved under transformation by $Q$ ($\|Qx\| = \|x\|$)
(D) All eigenvalues of $Q$ have an absolute magnitude equal to $1$
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
For an orthogonal matrix $Q$:
1. $Q^T Q = I \implies \det(Q^T Q) = (\det(Q))^2 = 1 \implies \det(Q) = \pm 1$. (A is true)
2. By definition of orthogonality, $Q^{-1} = Q^T$. (B is true)
3. $\|Qx\|^2 = (Qx)^T (Qx) = x^T Q^T Q x = x^T I x = \|x\|^2 \implies \|Qx\| = \|x\|$. (C is true)
4. If $\lambda$ is an eigenvalue with eigenvector $v$, $\|Qv\| = |\lambda| \|v\| = \|v\| \implies |\lambda| = 1$. (D is true)
Hence, all options A, B, C, and D are correct.