Linear Algebra Questions

If an n×n matrix A has fewer than n distinct eigenvalues, then A is not diagonalizable.

Every real 3×3| matrix must have a real eigenvalue.

Real eigenvalues of a real matrix always correspond to real eigenvectors.

Row operations on a matrix do not change its eigenvalues.

If A| is diagonalizable, then A| is invertible.

If v| is an eigenvector of A|, then cv| is also an eigenvector of A| for any number c that doesn't equal 0.

If the characteristic polynomial of a 2×2 matrix is λ2−5λ+6 then the determinant is 6.

If y is in the subspace W then the orthogonal projection of y onto W is y.

λ| is an eigenvalue of matrix A| if A−λI has linearly independent columns.

A 5×5| real matrix has an even number of real eigenvalues.

If A| is nxn and A| has n distinct eigenvalues, then the eigenvectors of A| are linearly independent.

If A is diagonalizable, then A^2 is also diagonalizable.

A matrix that is similar to the identity matrix is equal to the identity matrix.

A number c| is an eigenvalue of A| if and only if (A - cI)v = 0 has a nontrivial solution.

Suppose A is an 5×5 matrix. If A has three pivots, then ColA is a two-dimensional plane. (Yes/No/Maybe)

A| is invertible if and only 0 is not an eigenvalue of A|.

Matrices with the same eigenvalues are similar matrices.

The eigenvalues of A| are the entries on its main diagonal.

Cardinality of sets A and B if f is bijective

Cardinality of sets A and B if f is surjective