Question

Question: In Exercises 21 and 22, $A$ and $B$ are $n\times n$ matrices. Mark each statement True or False. Justify each answer.

1. If $A$ is $3\times 3$, with columns ${\mathrm{a}}_1$, ${\mathrm{a}}_2$, and ${\mathrm{a}}_3$, then $detA$ equals the volume of the parallelepiped determined by ${\mathrm{a}}_1$, ${\mathrm{a}}_2$, and ${\mathrm{a}}_3$.
2. $det{A}^T=\left(-1\right)detA$.
3. The multiplicity of a root $r$ of the characteristic equation of $A$ is called the algebraic multiplicity of $r$ as an eigenvalue of $A$.
4. A row replacement operation on $A$ does not change the eigenvalues.

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.

Step-by-step solution

Find an answer for part (a)

Assume that matrix $A$ be a $3\times 3$ matrix, then $\left|detA\right|$ is equal to the volume of the parallelepiped determined by the columns ${\mathrm{a}}_1$, ${\mathrm{a}}_2$, and ${\mathrm{a}}_3$ of $A$.

Therefore, the given statement is False.

Find an answer for part (b)

According to theorem 3 (Properties of Determinant): $det{A}^T=detA$.

Therefore, the given statement is False.

Find an answer for part (c)

The multiplicity of an eigenvalue $\lambda$ is its multiplicity as a root of the characteristic equation.

This implies that the multiplicity of a root $r$ of the characteristic equation of $A$ is called the algebraic multiplicity of $r$ as an eigenvalue of $A$

Therefore, the given statement is True.

Find an answer for part (d)

When we apply row replacement operation on the matrix $A$ , then the eigenvalues of the matrix $A$ change.

Therefore, the given statement is False.