Question

\(\mathrm{f}(\mathrm{x})=\left|\begin{array}{ccc}\cos \mathrm{x} & 0 & \sin \mathrm{x} \\ 0 & 1 & 0 \\ -\sin \mathrm{x} & 0 & \cos \mathrm{x}\end{array}\right|, \mathrm{g}(\mathrm{y})=\left|\begin{array}{ccc}\cos \mathrm{y} & -\sin \mathrm{y} & 0 \\ \sin \mathrm{y} & \cos \mathrm{y} & 0 \\ 0 & 0 & 1\end{array}\right|\) (i) \(\mathrm{f}(\mathrm{x}) \cdot \mathrm{f}(\mathrm{y})=\) (a) \(\mathrm{f}(\mathrm{xy})\) (b) \(\mathrm{f}(\mathrm{x} / \mathrm{y})\) (c) \(\mathrm{f}(\mathrm{x}+\mathrm{y})\) (d) \(\mathrm{f}(\mathrm{x}-\mathrm{y})\) (ii) Which of the following is correct? (a) \([\mathrm{f}(\mathrm{x})]^{-1}=[1 /\\{\mathrm{f}(\mathrm{x})\\}]\) (b) \([\mathrm{f}(\mathrm{x})]^{-1}=-\mathrm{f}(\mathrm{x})\) (c) \([\mathrm{f}(\mathrm{x})]^{-1}=\mathrm{f}(-\mathrm{x})\) (d) \([\mathrm{f}(\mathrm{x})]^{-1}=-\mathrm{f}(-\mathrm{x})\) (iii) \([\mathrm{f}(\mathrm{x}) \mathrm{g}(\mathrm{y})]^{-1}=\) (a) \(\mathrm{f}\left(\mathrm{x}^{-1}\right) \mathrm{g}\left(\mathrm{y}^{-1}\right)\) (b) \(\mathrm{f}\left(\mathrm{y}^{-1}\right) \mathrm{g}\left(\mathrm{x}^{-1}\right)\) (c) \(\mathrm{f}(-\mathrm{x}) \mathrm{g}(-\mathrm{y})\) (d) \(\mathrm{g}(-\mathrm{y}) \mathrm{f}(-\mathrm{x})\)

Short answer

(i) The correct answer is (c) \(f(x+y)\). (ii) The correct answer is (c) \(f(-x)\). (iii) The correct answer is (d) \(g(-y) f(-x)\).

Step-by-step solution

Compute the product f(x) * f(y)

To compute the product f(x) * f(y), we need to multiply their corresponding matrices: \[ \left(\begin{array}{ccc} \cos x & 0 & \sin x \\ 0 & 1 & 0 \\ -\sin x & 0 & \cos x \end{array}\right) \times \left(\begin{array}{ccc} \cos y & 0 & \sin y \\ 0 & 1 & 0 \\ -\sin y & 0 & \cos y \end{array}\right) \] By multiplying, we get: \[ \left(\begin{array}{ccc} \cos x \cos y-\sin x \sin y & 0 & \cos x \sin y+\sin x \cos y \\ 0 & 1 & 0 \\ -\sin x \cos y-\cos x \sin y & 0 & -\sin x \sin y+\cos x \cos y \end{array}\right) \] We recognize this as the matrix f(x+y), so the correct answer is (c) \(f(x+y)\).

Compute the inverse of f(x)

To compute the inverse of the matrix f(x): \[ \left(\begin{array}{ccc} \cos x & 0 & \sin x \\ 0 & 1 & 0 \\ -\sin x & 0 & \cos x \end{array}\right)^{-1} \] Since f(x) represents a rotation matrix in 3 dimensions, the inverse matrix is just its transpose: \[ \left(\begin{array}{ccc} \cos x & 0 & -\sin x \\ 0 & 1 & 0 \\ \sin x & 0 & \cos x \end{array}\right) \] This corresponds to the matrix f(-x), so the correct answer is (c) \(f(-x)\).

Compute the inverse of f(x) * g(y)

We want to compute the inverse of the matrix product f(x) * g(y): \[ \left(\begin{array}{ccc} \cos x & 0 & \sin x \\ 0 & 1 & 0 \\ -\sin x & 0 & \cos x \end{array}\right) \times \left(\begin{array}{ccc} \cos y & -\sin y & 0 \\ \sin y & \cos y & 0 \\ 0 & 0 & 1 \end{array}\right)^{-1} \] Recall that for a product of invertible matrices A and B, the inverse of their product is given by (AB)^{-1} = B^{-1} A^{-1}. Thus, we just have to compute the inverses we already found in the previous steps: \[ \left(g(y)^{-1}\right) \times \left(f(x)^{-1}\right) = \left(\begin{array}{ccc} \cos y & \sin y & 0 \\ -\sin y & \cos y & 0 \\ 0 & 0 & 1 \end{array}\right) \times \left(\begin{array}{ccc} \cos x & 0 & -\sin x \\ 0 & 1 & 0 \\ \sin x & 0 & \cos x \end{array}\right) \] As we can see, this corresponds to \(g(-y) f(-x)\), so the correct answer is (d) \(g(-y) f(-x)\).

Key concepts

Matrix Multiplication

When dealing with matrix multiplication, it's important to understand that the order of multiplication matters. Matrices are multiplied by taking the rows of the first matrix and columns of the second matrix and performing dot products. Each entry in the resulting matrix is obtained by summing the products of corresponding elements.

• Matrix multiplication is not commutative, meaning that \( AB eq BA \)
• For multiplication to be possible, the number of columns in the first matrix must equal the number of rows in the second
• The resulting matrix will have dimensions of the outer dimensions of the multiplied matrices

Understanding these basic rules can help you carry out operations accurately and identify patterns, like recognizing when a product gives you a specific transformation, such as a rotation by an angle in multivariable calculus.

Inverse Matrices

Inverse matrices play a key role in linear algebra, particularly when solving equations involving matrices. An inverse of a matrix \(A\), denoted \(A^{-1}\), is a matrix that, when multiplied by \(A\), results in the identity matrix. For square matrices, if the inverse exists, here are some important points to remember:

• A matrix has an inverse only if its determinant is non-zero
• The product of a matrix with its inverse is always the identity matrix, \(AA^{-1} = I\)
• For rotation matrices, which we'll cover soon, the inverse is often represented as the transpose

In the context of rotation matrices, the inverse can be particularly intuitive since it represents a rotation in the opposite direction.

Trigonometric Functions

Trigonometric functions help describe relationships in triangles and can be used to model periodic phenomena. In matrix operations, cosine and sine functions often appear in rotation matrices:

• \(\cos(\theta)\) and \(\sin(\theta)\) are key when defining the entries of a rotation matrix
• These functions also satisfy trigonometric identities like \( \cos^2(\theta) + \sin^2(\theta) = 1 \)
• Angles contribute to defining rotations in matrices by splitting into horizontal and vertical components

Understanding trigonometric functions is essential when interpreting matrices that involve circular or rotational transformations.

Rotation Matrix

A rotation matrix is a special kind of orthogonal matrix used primarily in 3D transformations to rotate objects in space. Here’s what makes them special:

• In 2D, a rotation matrix around the origin has the form \(\begin{bmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{bmatrix}\)
• In 3D, rotation matrices are extended to rotate around axes, such as the matrices given in the exercise
• The key property is that multiplication by a rotation matrix preserves length, meaning it preserves the magnitude and direction of the vectors

Rotation matrices are useful in computer graphics and robotics where spatial transformations are required.

3x3 Matrix

3x3 matrices are commonly used in various fields of science and engineering. They are particularly useful in three-dimensional (3D) computations. Characteristics of 3x3 matrices include:

• They can represent linear transformations in 3D space, such as rotations, scaling, and translations
• Applications include modeling systems in physics, engineering, and graphics
• Operations like finding the determinant, inverse, and eigenvalues are crucial for understanding their behavior

Working with 3x3 matrices effectively requires a good understanding of linear algebra concepts like matrix operations and the geometrical interpretations of these operations.