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Determinants and Cramer's Rule quiz

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Termini in questo insieme (15)

  • How do you calculate the determinant of a 2x2 matrix [[a, b], [c, d]]?
    Multiply a by d, multiply b by c, and subtract: ad - bc.
  • What is the determinant of the matrix [[8, 4], [5, 0]]?
    The determinant is 8*0 - 4*5 = -20.
  • What is the general formula for the determinant of a 2x2 matrix?
    The formula is ad - bc for a matrix [[a, b], [c, d]].
  • How do you handle negative numbers when calculating determinants?
    Multiply as usual and be careful with the signs; subtracting a negative is the same as adding.
  • What does Cramer's Rule allow you to do with systems of equations?
    Cramer's Rule lets you solve systems of equations by calculating determinants and plugging them into a formula.
  • When using Cramer's Rule for two equations, how do you set up the matrices for x and y?
    Replace the x or y column with the constants column to form new matrices for each variable.
  • What is the determinant of the matrix [[-3, 1], [-7, -2]]?
    The determinant is (-3)*(-2) - 1*(-7) = 6 + 7 = 13.
  • How do you calculate the determinant of a 3x3 matrix?
    Expand along the first row, multiplying each entry by the determinant of the 2x2 matrix left after removing its row and column, and alternate signs.
  • What is the sign pattern when expanding a 3x3 determinant along the first row?
    The signs alternate: +, -, +.
  • In Cramer's Rule for three variables, what does the denominator determinant (D) represent?
    It is the determinant of the coefficient matrix for x, y, and z.
  • How do you form the numerator determinant for x (Dx) in a 3x3 system using Cramer's Rule?
    Replace the x column in the coefficient matrix with the constants column, then calculate the determinant.
  • What do you do if the denominator determinant (D) is zero when using Cramer's Rule?
    If D is zero, the system has no unique solution.
  • How do you calculate the value of y using Cramer's Rule in a 3x3 system?
    Replace the y column with the constants column, calculate the determinant (Dy), and divide by D: y = Dy/D.
  • What is the solution to the system if Dx = -15 and D = -3?
    x = Dx/D = -15 / -3 = 5.
  • Why is it important to be careful with negative signs when calculating determinants?
    Because incorrect handling of negative signs can lead to wrong answers, especially when subtracting products.