Matrices Class Quiz

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Questions and Answers

What is the Cayley-Hamilton theorem primarily used for in relation to a matrix A?

To determine the rank of A
To express A in terms of its characteristic polynomial (correct)
To find the eigenvalues of A
To find the inverse of A

When applying the Cayley-Hamilton theorem to the matrix A = [[2, 5], [1, 4]], what is the first step?

Perform matrix diagonalization
Compute the characteristic polynomial of A (correct)
Calculate the determinant of A
Find the eigenvectors of A

To find the second derivative of the function z = x^3 + y^3 - 3axy, what must you do first?

Differentiate z with respect to y, then x
Apply the product rule to each term
Differentiate z with respect to x, then y
Find the first order partial derivatives first (correct)

What is the nth derivative of the function log(x + 3)?

(-1)^(n-1)/(x+3)^n (B) Signup and view all the answers

If x = r cos(θ) and y = r sin(θ), what does the relationship (∂x/∂r) + (∂y/∂r) equal?

1 (D) Signup and view all the answers

Verify that the mixed partial derivatives satisfy what relation for the function u = x^3 + y^3 - 3axy?

∂²u/∂x∂y = ∂²u/∂x² (A) Signup and view all the answers

If u = sin⁻¹(y/x), what is the derivative ∂u/∂x?

y/(√(x² - y²)) (C) Signup and view all the answers

What is the relationship defined by Euler's theorem for a homogeneous function of degree n?

$2 \frac{\partial^2 u}{\partial x^2} + 2xy \frac{\partial^2 u}{\partial x \partial y} + 2 \frac{\partial^2 u}{\partial y^2} = n(n-1)u$ (C) Signup and view all the answers

When using polar coordinates, what are the expressions for x and y if x = r cos θ and y = r sin θ?

x = r cos θ and y = r sin θ (B) Signup and view all the answers

What is the result of the expression $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y}$ when u = log( (x+y)/(x-y))?

3 (A) Signup and view all the answers

In the context of multiple variables, what does the Jacobian determinant represent?

The rate of change of all variables collectively (A) Signup and view all the answers

For the function u = xy^2, what is the result of the mixed partial derivative ∂²u/∂x∂y?

2y (A) Signup and view all the answers

Which of the following expressions defines the relationship for homogenous functions in terms of their variables?

For all constants c, u(cx, cy) = c^n u(x, y) (D) Signup and view all the answers

In the expression ∂(u,v,w)/∂(x,y,z) for u = xyz, the determinant gives which of the following quantities?

The volume in 3D space formed by vectors (B) Signup and view all the answers

What pattern can be observed when evaluating an integral that involves sine and cosine over a defined interval?

Complete cancellation resulting in zero (B) Signup and view all the answers

In the process of evaluating the integral of sin^5(x) in the interval from 0 to π/2, what key aspect is considered?

The power reduction technique for simplification (D) Signup and view all the answers

If $y = a ext{cos}( ext{log} x) + b ext{sin}( ext{log} x)$, what is the form of the equation when finding the derivative?

$x^2 y_{n+2} + (2n + 1) x y_{n+1} + (n^2 + 1) y_n = 0$ (B) Signup and view all the answers

What result occurs when applying Leibniz's Theorem to $x e^y$?

The nth derivative can always be expressed in terms of the lower derivatives. (A) Signup and view all the answers

For $y = ext{sin}^{-1}(x)$, which expression correctly reflects the nth derivative?

$(1 + x^2) y_{n+2} + (2n + 1) x y_{n+1} + n^2 y_n = 0$ (C) Signup and view all the answers

If $u = ext{log}(rac{x+y}{x-y})$, what must be shown about the derivatives in relation to $x$ and $y$?

$x u_x + y u_y = 1$ (C) Signup and view all the answers

What does the expression $z = x^2 an^{-1}(x) - y^2 an^{-1}(y)$ express?

The relationship between partial derivatives and the coordinates. (A) Signup and view all the answers

When deriving $y = ext{log}(x + ext{sqrt}(1 + x^2))$, which relationship is expected?

$(1 + x^2) y_{n+2} + (2n + 1) x y_{n+1} + n^2 y_n = 0$ (D) Signup and view all the answers

If $cos^{-1}(b) = ext{log}(n)$, which equation must hold true for the derivatives?

$x^2 y_{n+2} + (2n + 1) x y_{n+1} + n^2 y_n = 0$ (A) Signup and view all the answers

If $u = ext{sin}^{-1}(rac{x}{x+y})$, what proves the relationship between derivatives of u with respect to x and y?

$x u_x + y u_y = 1$ (A) Signup and view all the answers

Flashcards

Homogeneous function of degree n

A function where replacing x and y with kx and ky respectively results in the function being multiplied by k^n

Euler's theorem for homogeneous functions

A theorem relating the sum of partial derivatives of a homogeneous function to the function itself

Partial derivatives of homogeneous functions

Derivatives of a function with respect to one variable, holding others constant

Evaluate definite integral of trigonometric functions

Calculate numerical value definite integral of trigonometric functions like sin^m x cos^n x using trigonometric identities