Understanding Determinants

What is a determinant in linear algebra?

• A determinant is a vector that represents the direction of a matrix.

• A determinant is a matrix that contains only zeros.

• A determinant is a scalar value that indicates the invertibility of a square matrix.

• A determinant is a function that maps matrices to real numbers.

How do you calculate the determinant of a 2x2 matrix?

• determinant = a * b * c * d

• determinant = (a + d) + (b + c)

• determinant = (a * d) - (b * c)

• determinant = (a - d) + (b * c)

What is the geometric interpretation of a determinant?

• The determinant indicates the angle between vectors in a matrix.

• The determinant represents the volume scaling factor of the transformation defined by the matrix.

• The determinant measures the distance between points in space.

• The determinant is the sum of the elements in the matrix.

What is the determinant of a matrix with a row of zeros?

• 0

• 1

• -1

• 5

How does the determinant change when two rows of a matrix are swapped?

• The determinant doubles in value.

• The determinant remains unchanged.

• The determinant becomes zero.

• The determinant changes sign.

What is the relationship between the determinant and the invertibility of a matrix?

• The determinant of a matrix indicates its invertibility; a non-zero determinant means the matrix is invertible.

• A non-zero determinant indicates the matrix is singular.

• The determinant has no relation to the invertibility of a matrix.

• A matrix is always invertible if its determinant is zero.

How do you find the determinant of a 3x3 matrix using cofactor expansion?

• The determinant can be found by taking the trace of the matrix.

• The determinant of a 3x3 matrix A = [[a, b, c], [d, e, f], [g, h, i]] is calculated as: det(A) = a(ei - fh) - b(di - fg) + c(dh - eg).

• The determinant is found by adding all elements of the matrix.

• The determinant is calculated by multiplying all diagonal elements together.

What is the effect of multiplying a row of a matrix by a scalar on its determinant?

• The determinant is divided by the scalar.

• The determinant is squared.

• The determinant remains unchanged.

• The determinant is multiplied by the scalar.

Can the determinant of a matrix be negative?

• No, the determinant can only be positive.

• The determinant can only be a whole number.

• The determinant is always zero.

• Yes, the determinant of a matrix can be negative.

What is the determinant of the identity matrix?

• -1

• 1

• 2

• 0