Q1286: If $A=\begin{bmatrix}1&cot~x\\ -cot~x&1\end{bmatrix}$ show t...

Class CBSE Class 12 Mathematics Matrices and Determinants Q #1286

KNOWLEDGE BASED

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5 Marks 2024 AISSCE(Board Exam) LA

If $A=\begin{bmatrix}1&cot~x\\ -cot~x&1\end{bmatrix}$ show that $A^{\prime}A^{-1}=\begin{bmatrix}-cos~2x&-sin~2x\\ sin~2x&-cos~2x\end{bmatrix}$

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AI Tutor Explanation

Detailed Solution

Step 1: Find $A'$

Given $A=\begin{bmatrix}1&cot~x\\ -cot~x&1\end{bmatrix}$, the transpose $A'$ is obtained by interchanging rows and columns: $$A' = \begin{bmatrix}1&-cot~x\\ cot~x&1\end{bmatrix}$$

Step 2: Find $|A|$

The determinant of $A$ is: $$|A| = (1)(1) - (cot~x)(-cot~x) = 1 + cot^2x = cosec^2x$$

Step 3: Find $A^{-1}$

The inverse of $A$ is given by: $$A^{-1} = \frac{1}{|A|} adj(A) = \frac{1}{cosec^2x} \begin{bmatrix}1&-cot~x\\ cot~x&1\end{bmatrix} = sin^2x \begin{bmatrix}1&-cot~x\\ cot~x&1\end{bmatrix} = \begin{bmatrix}sin^2x&-sin^2x~cot~x\\ sin^2x~cot~x&sin^2x\end{bmatrix} = \begin{bmatrix}sin^2x&-sinx~cosx\\ sinx~cosx&sin^2x\end{bmatrix}$$

Step 4: Find $A'A^{-1}$

Now, we compute $A'A^{-1}$: $$A'A^{-1} = \begin{bmatrix}1&-cot~x\\ cot~x&1\end{bmatrix} \begin{bmatrix}sin^2x&-sinx~cosx\\ sinx~cosx&sin^2x\end{bmatrix} = \begin{bmatrix}sin^2x - cotx~sinx~cosx & -sinx~cosx - cotx~sin^2x\\ cotx~sin^2x + sinx~cosx & -cotx~sinx~cosx + sin^2x\end{bmatrix} = \begin{bmatrix}sin^2x - cos^2x & -2sinx~cosx\\ 2sinx~cosx & sin^2x - cos^2x\end{bmatrix} = \begin{bmatrix}-cos~2x&-sin~2x\\ sin~2x&-cos~2x\end{bmatrix}$$

Final Answer: $A^{\prime}A^{-1}=\begin{bmatrix}-cos~2x&-sin~2x\\ sin~2x&-cos~2x\end{bmatrix}$

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Pedagogical Audit

Bloom's Analysis: This is an REMEMBER question because it requires recalling the definitions of transpose and inverse of a matrix and applying trigonometric identities.

Knowledge Dimension: PROCEDURAL

Justification: The question involves applying a series of steps to find the transpose, inverse, and product of matrices, which falls under procedural knowledge.<\/span>

Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. It directly tests the student's understanding of matrix operations and trigonometric identities as covered in the textbook.