Question

The determinant of an orthogonal matrix is:

• A. \(\pm 1\)
• B. \(2\)
• C. \(0\)
• D. \(\pm 2\)

Hint:

For orthogonal matrices: [ A^T A = I quad Rightarrow quad det(A) = pm 1 ] This property is frequently tested in exams.

Answer 1

The Correct Option is A

Step 1: Recall the definition of an orthogonal matrix. A square matrix \(A\) is said to be orthogonal if: \[ A^T A = I \] where \(A^T\) is the transpose of \(A\) and \(I\) is the identity matrix. Step 2: Take determinant on both sides. \[ \det(A^T A) = \det(I) \] Step 3: Use properties of determinants. \[ \det(A^T A) = \det(A^T)\det(A) \] \[ \det(A^T) = \det(A) \] So, \[ \det(A)^2 = 1 \] Step 4: Solve for \(\det(A)\). \[ \det(A) = \pm 1 \] Step 5: Final conclusion. The determinant of an orthogonal matrix is always \(\boxed{\pm 1}\).