Question Bank • QB_EM_LA_005 NAT • 1 Mark

Section 1: Engineering Mathematics

Linear Algebra

Subtopic: Linear and orthogonal transformations

Question Statement:

Let $Q$ be a $3 \times 3$ real orthogonal matrix. The determinant of the matrix $(Q^T Q + 3I)$ is ________ (answer in integer).

Answer Key:

64

Step-by-Step Solution & Explanation

For any real orthogonal matrix $Q$, we have $Q^T Q = I$. Therefore: $Q^T Q + 3I = I + 3I = 4I$ Taking the determinant for a $3 \times 3$ matrix: $\det(4I) = 4^3 \det(I) = 64 \times 1 = 64$
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