If \( A = \begin{bmatrix} \sqrt{2} & 1 \\ -1 & \sqrt{2} \end{bmatrix} \), \( B = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} \), \( C = ABA^\top \) and \( X = A^\top C^2 A \), then \( \det (X) \) is equal to:
- 243
- 729
- 27
- 891
The Correct Option is B
Approach Solution - 1
Given:
\( A = \begin{bmatrix} \sqrt{2} & 1 \\ -1 & \sqrt{2} \end{bmatrix} \), \( B = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \)
First, find:
\( \text{det}(A) = (\sqrt{2}) \times (\sqrt{2}) - (1) \times (-1) = 3 \)
\( \text{det}(B) = 1 \)
Now, compute \( C = ABA^T \). Since \( \text{det}(C) = (\text{det}(A))^2 \times \text{det}(B) \):
\( \text{det}(C) = 3^2 \times 1 = 9 \)
For \( X = A^T C^2 A \), we use:
\( \text{det}(X) = [\text{det}(A^T)] \times [\text{det}(C^2)] \times [\text{det}(A)] \)
Since \( \text{det}(A^T) = \text{det}(A) \) and \( \text{det}(C^2) = (\text{det}(C))^2 \):
\( \text{det}(X) = (\text{det}(A)) \times (\text{det}(C))^2 \times (\text{det}(A)) \)
\( \text{det}(X) = 3 \times 9^2 \times 3 = 729 \)
Approach Solution -2
Given matrices \( A = \begin{bmatrix} \sqrt{2} & 1 \\ -1 & \sqrt{2} \end{bmatrix} \), \( B = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} \), \( C = ABA^\top \) and \( X = A^\top C^2 A \), find \( \det(X) \).
Concept Used:
Properties of determinants:
- \( \det(AB) = \det(A)\det(B) \)
- \( \det(A^\top) = \det(A) \)
- \( \det(A^n) = (\det(A))^n \)
Step-by-Step Solution:
Step 1: Compute \( \det(A) \).
\[ \det(A) = (\sqrt{2})(\sqrt{2}) - (1)(-1) = 2 + 1 = 3 \]Step 2: Compute \( \det(B) \).
\[ \det(B) = (1)(1) - (0)(1) = 1 \]Step 3: Express \( \det(X) \) in terms of known determinants.
Given \( C = ABA^\top \), and \( X = A^\top C^2 A \).
\[ \det(X) = \det(A^\top C^2 A) = \det(A^\top) \cdot \det(C^2) \cdot \det(A) \] \[ = \det(A) \cdot (\det(C))^2 \cdot \det(A) = (\det(A))^2 \cdot (\det(C))^2 \]Step 4: Compute \( \det(C) \).
\[ C = ABA^\top \Rightarrow \det(C) = \det(A) \cdot \det(B) \cdot \det(A^\top) \] \[ = \det(A) \cdot \det(B) \cdot \det(A) = (\det(A))^2 \cdot \det(B) \] \[ = (3)^2 \cdot 1 = 9 \]Step 5: Compute \( \det(X) \).
\[ \det(X) = (\det(A))^2 \cdot (\det(C))^2 = (3)^2 \cdot (9)^2 = 9 \cdot 81 = 729 \]Therefore, \( \det(X) = \mathbf{729} \).
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