Understanding Diagonalizable Matrices

What is a necessary condition for an N by N matrix to be diagonalizable?

• It must be a square matrix.

• It must be a symmetric matrix.

• It must have n distinct eigenvalues.

• It must have a determinant of zero.

What does the geometric multiplicity of an eigenvalue represent?

• The number of times the eigenvalue appears in the characteristic polynomial.

• The dimension of the eigenspace corresponding to the eigenvalue.

• The determinant of the matrix.

• The trace of the matrix.

How is the algebraic multiplicity of an eigenvalue determined?

• By the trace of the matrix.

• By the number of times the eigenvalue appears in the characteristic polynomial.

• By the determinant of the matrix.

• By the number of linearly independent eigenvectors.

In the given 3x3 matrix example, what is the algebraic multiplicity of the eigenvalue 3?

• 0

• 2

• 1

• 3

What is the geometric multiplicity of the eigenvalue 5 in the 3x3 matrix example?

• 1

• 2

• 3

• 0

What does the theorem state about the relationship between geometric and algebraic multiplicities?

• Geometric multiplicity is always less than algebraic multiplicity.

• Geometric multiplicity is always equal to algebraic multiplicity.

• Geometric multiplicity is always greater than algebraic multiplicity.

• Geometric multiplicity is always less than or equal to algebraic multiplicity.

Under what condition is a square matrix diagonalizable according to the theorem?

• If the geometric multiplicity of every eigenvalue is equal to its algebraic multiplicity.

• If the matrix is symmetric.

• If the determinant is non-zero.

• If the trace is zero.

What is the sum of the algebraic multiplicities of a matrix equal to?

• The degree of the characteristic polynomial.

• The trace of the matrix.

• The number of distinct eigenvalues.

• The determinant of the matrix.

What is the significance of having enough eigenvectors in a matrix?

• It ensures the matrix is invertible.

• It ensures the matrix has a zero determinant.

• It ensures the matrix is symmetric.

• It ensures the matrix is diagonalizable.

What additional resources are suggested for further learning?

• A video on matrix inversion.

• A video on linear transformations.

• A video on eigenvalues and eigenvectors.

• A video on matrix multiplication.