Homepage › Solution manuals › Stephen Friedberg › Linear Algebra › Exercise 4.5.1

Linear Algebra

Exercise 4.5.1

Answers

1.

No. For example, we have

\det (2 I) \neq 2 \det (I)

when

I

is the

2 \times 2

identity matrix.

2.

Yes. This is Theorem 4.3.

3.

Yes. This is Theorem 4.10.

4.

No. Usually it should be

δ (B) = - δ (A)

.

5.

No. Both the determinant and the zero function,

δ (A) = 0

, are

n

-linear function.

6.

Yes. Let

v_{1}, u_{1}, u_{2}, \dots, u_{n}

are row vectors and

c

are scalar. We have

δ (\begin{matrix} u_{1} + c v_{1} \\ u_{2} \\ ⋮ \\ u_{n} \end{matrix}) = 0 = 0 + c \cdot 0 = δ (\begin{matrix} u_{1} \\ u_{2} \\ ⋮ \\ u_{n} \end{matrix}) + cδ (\begin{matrix} v_{1} \\ u_{2} \\ ⋮ \\ u_{n} \end{matrix}) .

The cases for other rows are similar.

jephianlin

2011-06-27 00:00

Comments