Exercise 4.3.1

Answers

1.: No. The elementary of type 2 has determinant other than $1$ .
2.: Yes. This is Theorem 4.7.
3.: No. A matrix is invertible if and only if its determinant is not zero.
4.: Yes. The fact that $n \times n$ matrix $A$ has rank $n$ is equivalent to the fact that $A$ is invertible and the fact that $\det (A) \neq 0$ .
5.: No. We have that $\det (A^{t}) = \det (A)$ by Theorem 4.8.
6.: Yes. This is the instant result of Theorem 4.4 and Theorem 4.8.
7.: No. It still require the condition that the determinant cannot be zero.
8.: No. The matrix $M_{k}$ is the matrix obtained from $A$ by replacing column $k$ of $A$ by $b$ .

jephianlin

2011-06-27 00:00