Question:
If $A = \begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}$, then $\text{adj}\ A =$
If $A = \begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}$, then $\text{adj}\ A =$
Show Hint
Recognize that the upper-left $2 \times 2$ block of this matrix is a standard rotation matrix. Rotation matrices are orthogonal, meaning their inverse and adjoint are equal to their transpose.
To transpose a rotation matrix, keep the diagonal intact and simply flip the signs of the off-diagonal symmetric elements!
To transpose a rotation matrix, keep the diagonal intact and simply flip the signs of the off-diagonal symmetric elements!
Updated On: Jun 4, 2026
- $\begin{bmatrix} -\cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}$
- $\begin{bmatrix} \cos\theta & \sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}$
- $\begin{bmatrix} \cos\theta & \sin\theta & 0 \\ -\sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}$
- $\begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}$
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Question:
We are given a $3 \times 3$ matrix $A$ representing a rotation about the $z$-axis. We need to determine its classical adjoint matrix, denoted as $\text{adj}\ A$.
Step 2: Key Formula or Approach:
The matrix $A$ given here is an orthogonal matrix. For any standard orthogonal matrix, the inverse matrix satisfies $A^{-1} = A^T$ (the transpose of $A$).
Additionally, we know the matrix identity relating the inverse and the adjoint:
$$A^{-1} = \frac{\text{adj}\ A}{|A|}$$ Let's compute the determinant $|A|$ first. If $|A| = 1$, then $\text{adj}\ A = A^T$, which saves us from computing nine separate cofactor elements manually.
Step 3: Detailed Explanation:
Let's find the determinant of $A$ by expanding along the third row or third column:
$$|A| = 1 \cdot (\cos^2\theta - (-\sin^2\theta)) = \cos^2\theta + \sin^2\theta = 1$$ Since $|A| = 1$, we can use our orthogonal property shortcut:
$$\text{adj}\ A = A^T$$ Let's find the transpose of matrix $A$ by swapping its rows and columns:
$$A^T = \begin{bmatrix} \cos\theta & \sin\theta & 0 \\ -\sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ Therefore, the adjoint matrix is simply the transpose of the initial matrix.
Step 4: Final Answer:
The matrix matching $\text{adj}\ A$ is represented by option (C).
We are given a $3 \times 3$ matrix $A$ representing a rotation about the $z$-axis. We need to determine its classical adjoint matrix, denoted as $\text{adj}\ A$.
Step 2: Key Formula or Approach:
The matrix $A$ given here is an orthogonal matrix. For any standard orthogonal matrix, the inverse matrix satisfies $A^{-1} = A^T$ (the transpose of $A$).
Additionally, we know the matrix identity relating the inverse and the adjoint:
$$A^{-1} = \frac{\text{adj}\ A}{|A|}$$ Let's compute the determinant $|A|$ first. If $|A| = 1$, then $\text{adj}\ A = A^T$, which saves us from computing nine separate cofactor elements manually.
Step 3: Detailed Explanation:
Let's find the determinant of $A$ by expanding along the third row or third column:
$$|A| = 1 \cdot (\cos^2\theta - (-\sin^2\theta)) = \cos^2\theta + \sin^2\theta = 1$$ Since $|A| = 1$, we can use our orthogonal property shortcut:
$$\text{adj}\ A = A^T$$ Let's find the transpose of matrix $A$ by swapping its rows and columns:
$$A^T = \begin{bmatrix} \cos\theta & \sin\theta & 0 \\ -\sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ Therefore, the adjoint matrix is simply the transpose of the initial matrix.
Step 4: Final Answer:
The matrix matching $\text{adj}\ A$ is represented by option (C).
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