Equations & Eigenvalues

Solve the equation: 2x + 5 = 11

• x = 3

• x = 4

• x = 6

• x = 5

Find the eigenvalues of the matrix: [[3, 1], [1, 3]]

• [3, 3]

• [2, 2]

• [4, 2]

• [5, 1]

Solve the system of equations: 3x + 2y = 8, 2x - y = 1

• x = 2, y = 3

• x = 3, y = 4

• x = 1, y = 5

• x = 4, y = 2

What are eigenvectors and why are they important in linear algebra?

• Eigenvectors are only applicable to non-linear transformations.

• Eigenvectors are vectors that do not change direction when a linear transformation is applied to them, only their magnitude can change. They are important in linear algebra because they help in understanding the behavior of linear transformations and diagonalizing matrices.

• Eigenvectors are vectors that change direction when a linear transformation is applied to them.

• Eigenvectors are not important in linear algebra and are rarely used.

Apply matrix operation: [[1, 2], [3, 4]] + [[-1, 0], [2, 1]]

• [[0, 2], [5, 5]]

• [[1, 2], [3, 4]] + [[-1, 0], [2, 1]] = [[2, 2], [5, 5]]

• [[1, 2], [3, 4]] + [[-1, 0], [2, 1]] = [[0, 2], [1, 5]]

• [[1, 2], [3, 4]] + [[-1, 0], [2, 1]] = [[-1, 2], [3, 5]]

Interpret geometrically the solution of the equation: 2x - 3y = 6

• The solution of the equation 2x - 3y = 6 can be interpreted geometrically as a line with a slope of 2/3 passing through the point (0, -2) on the y-axis.

• The solution of the equation 2x - 3y = 6 is a straight line passing through the point (0, 0).

• The solution of the equation 2x - 3y = 6 is a parabola opening downwards.

• The solution of the equation 2x - 3y = 6 represents a circle with a radius of 6 centered at the origin.

Solve the equation: 4y - 7 = 5

• y = 6

• y = 3

• y = 4

• y = 2

Determine the eigenvalues of the matrix: [[2, 0], [0, 2]]

• [2, 2]

• [1, 1]

• [0, 0]

• [3, 3]

Find the solution to the system of equations: x + y = 5, 2x - y = 3

• x = 1, y = 4

• x = 2, y = 3

• x = 3/2, y = 7/2

• x = 8/3, y = 7/3

Explain the significance of eigenvalues in diagonalizing matrices.

• Eigenvalues determine the rank of the matrix

• Eigenvalues are only applicable to square matrices

• Eigenvalues are used to calculate determinants of matrices

• Eigenvalues provide the scaling factor for eigenvectors, simplifying matrix calculations.