Exercise 3.1.1

Answers

1.

Yes. Since every elementary matrix comes from

I_{n}

, a square matrix.

2.

No. For example,

2 I_{1}

is an elementary matrix of type 2.

3.

Yes. It’s an elementary matrix of type 2 with scalar

1

4.

No. For example, the product of two elementary matrices

(\begin{matrix} 2 & 0 \\ 0 & 1 \end{matrix}) (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}) = (\begin{matrix} 0 & 2 \\ 1 & 0 \end{matrix})

is not an elementary matrix.

5.

Yes. This is Theorem 3.2.

6.

No. For example, the sum of two elementary matrices

(\begin{matrix} 2 & 0 \\ 0 & 1 \end{matrix}) (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}) = (\begin{matrix} 2 & 1 \\ 1 & 1 \end{matrix})

is not an elementary matrix.

7.

Yes. See Exercise 3.1.5.

8.

No. For example, let

A = (\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix})

and

B = (\begin{matrix} 1 & 0 \\ 1 & 0 \end{matrix})

. Then we can obtain

B

by add one time the first row of

A

to the second row of

B

. But all column operation on

A

can not change the fact that the second row of

A

is two zeros.

9.

Yes. If

B = EA

, we have

E^{- 1} B = A

and

E^{- 1}

is an elementary matrix of row operation.

jephianlin

2011-06-27 00:00