Question

If $A = \begin{bmatrix} \cos \theta & \sin \theta & 0 \\ -\sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix}$, where $A_{21}, A_{22}, A_{23}$ are cofactors of $a_{21}, a_{22}, a_{23}$ respectively, then the value of $a_{21}\text{A}_{21} + \text{a}_{22}\text{A}_{22} + \text{a}_{23}\text{A}_{23} =$

• A. 1
• B. -1
• C. 0
• D. 2

Hint:

Row $\times$ cofactors = determinant.

Answer 1

The Correct Option is A

Concept:
Sum of elements of any row multiplied by their cofactors equals determinant of matrix: \[ a_{i1}A_{i1} + a_{i2}A_{i2} + a_{i3}A_{i3} = |A| \] Step 1: Identify row. \[ a_{21}A_{21} + a_{22}A_{22} + a_{23}A_{23} = |A| \]
Step 2: Find determinant. \[ |A| = \begin{vmatrix} \cos\theta & \sin\theta & 0 \\ -\sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{vmatrix} \] \[ = 1 \cdot \begin{vmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{vmatrix} \] \[ = \cos^2\theta + \sin^2\theta = 1 \]
Step 3: Conclusion. \[ a_{21}A_{21} + a_{22}A_{22} + a_{23}A_{23} = 1 \]