Visual SolutionPYQ 2024 · Jan 27 Shift 1Easy

Question

Consider the matrix

f(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}

.\n\nStatement I:

f(-x)

is the inverse of

f(x)

.\nStatement II:

f(x) \cdot f(y) = f(x + y)

.\n\nWhich of the following is correct?

(A)Statement I is false but Statement II is true

(B)Both Statement I and Statement II are false

(C)Statement I is true but Statement II is false

(D)Both Statement I and Statement II are true

Solution Path

f(x)

is a rotation matrix.

f(-x) = f(x)^{-1}

via

\cos^2+\sin^2=1

f(x)f(y) = f(x+y)

via angle addition. Both true.

01Question Setup

1/4

Rotation matrix

f(x)

3 \times 3

with

\cos x, -\sin x

in the top-left block. Check two statements about its properties.

Check Statements I and II

02Statement I: Inverse

2/4

Compute

f(x) \cdot f(-x)

. Using

\cos^2 x + \sin^2 x = 1

, the product equals the identity matrix

I

f(x) \cdot f(-x) = I \implies f(-x) = f(x)^{-1}

- TRUE

03Statement II: CompositionKEY INSIGHT

3/4

Rotation composition = angle addition. By the angle addition formulas,

f(x) \cdot f(y) = f(x+y)

f(x) \cdot f(y) = f(x+y)

- TRUE

04Final Answer

4/4

Both statements are true.

f(x)

is an orthogonal rotation matrix.

\boxed{\text{Both true}}

- Answer (D)

Concepts from this question1 concepts unlocked

EASY

For a square matrix A: tr(A) = sum of eigenvalues, det(A) = product of eigenvalues.

\text{tr}(A) = \sum \lambda_i, \quad |A| = \prod \lambda_i

Converts matrix questions into simple algebra on eigenvalues. Saves you from computing A² explicitly.

Trace problemsDeterminant shortcutsCharacteristic equation

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