Question

Question: In Exercises 21 and 22, A is an $n\times n$ matrix. Mark each statement True or False. Justify each answer.

1. If$A\mathbf{x}=\lambda \mathbf{x}$for some vector x, then$x$is an eignvector of A.
2. If${\mathbf{v}}_{\mathbf{1}}$and${\mathbf{v}}_{\mathbf{2}}$are linearly independent eigenvectors, then they corresponds to distinct values.
3. A steady-state vector for a stochastic matrix is actually an eigenvector.
4. The eigenvalues of a matrix are on its main diagonal.
5. An eigenspace of A is a null space of a certain matrix.

Short answer

a. The given statement is False.

b. The given statement is False.

c. The given statement is True

d. The given statement is False.

e. The given statement is True.

Step-by-step solution

Find an answer for part (a)

The vector x in the equation$A\mathbf{x}=\lambda \mathbf{x}$ must be non-zero, then only $\lambda$ will be the eigenvalue of A.

Thus, statement (a) is false.

Find an answer for part (b)

According to Theorem 2, if $v_1$,……, $v_r$ are eigenvectors, corresponds to distinct eigenvalues ${\lambda }_1$,…..,${\lambda }_r$ of a matrix of order $n\times n$, then the set of vectors is linearly independent.

The Eigenvectors corresponding to Eigenvalues may or may not be linearly independent.

Thus, statement (b) is false.

Find an answer for part (c)

If A is a Stochastic matrix, then for a vector q, the equation is $A\mathbf{q}=\mathbf{q}$.

Thus, the steady-state vector for a stochastic matrix is an eigenvector.

Thus, statement (c) is true.

Find an answer for part (d)

According to Theorem 1, If the eigenvalues of the matrix are in its main diagonal, then the matrix is a triangular matrix.

Thus, statement (d) is false.

Find the answer for part (e)

If xis the eigenvector of a matrix A, then;

$\begin{array}{c}A\mathbf{x}=\lambda \mathbf{x}\\ \left(A-\lambda I\right)\mathbf{x}=0\end{array}$

The eigenvectors are the non-trivial solution of the system of equations.

Thus, statement (e) is true.