Matrices & Determinants
Orthogonal Matrices and Determinants
Grade 12
Question:
<p>If A and B are two orthogonal matrices of order n and \(\det(A) + \det(B) = 0\), then which of the following must be correct?</p>
<p>(a) \(\det(A + B) = \det(A) + \det(B)\)</p>
<p>(b) \(\det(A + B) = 0\)</p>
<p>(c) A and B both are singular matrices</p>
<p>(d) \(A + B = 0\)</p>
Step-by-Step Solution
Key Concept: Orthogonal matrices have determinants of ±1. Use the constraint on determinants to determine which properties must hold.
<p>For orthogonal matrices, $\det(A) = \pm 1$ and $\det(B) = \pm 1$. Given $\det(A) + \det(B) = 0$, we have either $\det(A) = 1, \det(B) = -1$ or $\det(A) = -1, \det(B) = 1$.</p><p>Since orthogonal matrices are invertible, neither A nor B is singular, eliminating (c). We can verify (a) and (b) hold under these conditions.</p>
Correct Answer: a, b