Mathematics: Matrix Equation and Inverse Matrix
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Questions and Answers

What is the sum of a symmetric matrix and a skew-symmetric matrix?

  • A zero matrix (correct)
  • A matrix that is only skew-symmetric
  • An identity matrix
  • A matrix that is only symmetric
  • What is the derivative of $\sin(x)\sin(x)$ with respect to $x$?

  • $2\sin(x)\cos(x)$
  • $\sin^2(x)$
  • $\cos^2(x)$
  • $2\cos(x)\sin(x)$ (correct)
  • For $x = a(\theta + \sin(\theta))$ and $y = a(1 - \cos(\theta))$, what is $dx$ in terms of trigonometric functions?

  • $\sec^2(\theta)$
  • $\tan(2\theta)$ (correct)
  • $-a\sin(2\theta)$
  • $a\cos(2\theta)$
  • In which intervals is the function $f(x) = 10 - 6x - 2x^2$ decreasing?

    <p>$(-\infty, -1)$ and $(1, \infty)$</p> Signup and view all the answers

    What is the result of evaluating $\int (x-1)(x-2) dx$?

    <p>$x^3 - 3x^2 + 4x$</p> Signup and view all the answers

    What is the area of the parallelogram formed by vectors $\mathbf{a} = \mathbf{i} + \mathbf{j} - \mathbf{k}$ and $\mathbf{b} = \mathbf{i} - \mathbf{j} + \mathbf{k}$?

    <p>$8$</p> Signup and view all the answers

    Which of the following correctly represents the inverse of the product of invertible matrices A and B?

    <p>(AB)^{-1} = A^{-1}B^{-1}</p> Signup and view all the answers

    Given x = a(θ - sinθ) and y = a(1 + cosθ), what is dx?

    <p>dx = -a cos(θ)</p> Signup and view all the answers

    For the function f(x) = x^2 - 4x + 6, in which interval is f decreasing?

    <p>(2, ∞)</p> Signup and view all the answers

    What is the integral of (x+1)(x+2) dx?

    <p>(x^2 + 3x + 4) + C</p> Signup and view all the answers

    Given sec^2(x) tan(y) dx + sec^2(y) tan(x) dy = 0, solve the differential equation.

    <p>y = x</p> Signup and view all the answers

    If one defective item is randomly chosen from a stockpile containing items from machines A and B, what is the probability it was produced by machine B?

    <p>0.40</p> Signup and view all the answers

    If the function 𝑓(𝑥) = 𝑎𝑥 + 𝑏$ is continuous for $2 < x < 10$, what are the values of a and b?

    <p>a = 1, b = 0</p> Signup and view all the answers

    Using the equation 𝐴² - 𝐴 + 2𝐼 = 𝑂, find the inverse of matrix A if 𝐴 = $\begin{bmatrix}4 & -2 \ 4 & -2 \end{bmatrix}$, 𝐼 is 2x2 identity matrix, and 𝑂 is the zero matrix.

    <p>$\begin{bmatrix}0.5 &amp; 0.5 \ -1 &amp; 2 \end{bmatrix}$</p> Signup and view all the answers

    In the relation 𝑅 = {(1,1), (2,2), (3,3), (1,3)}, which ordered pair should be added to make R the smallest equivalence relation?

    <p>(2,1)</p> Signup and view all the answers

    Given the function 𝑓: 𝑅 → 𝑅 as 𝑓(𝑥) = |𝑥| for 𝑥 ∈ 𝑅, what type of function is 𝑓?

    <p>Neither one-one nor onto function</p> Signup and view all the answers

    If a function is both one-one and onto, what type of function is it?

    <p>Bijection</p> Signup and view all the answers

    What is the result when the determinant of a matrix is zero?

    <p>The matrix is non-invertible</p> Signup and view all the answers

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