Exercise 5.3.8

There’s no better method to check whether the matrix is regular or not. So just try it or find the evidence that the power of it will not be a positive matrix. When the matrix is nonnegative, we may just consider the matrix obtained by replacing each nonzero entry by $1$.

1.

Yes. The square of it is positive.

2.

Yes. It’s positive when the power is $4$.

3.

No. The second and the third column always interchange each time.

4.

No. The second column does not change each time.

5.

No. The second and the third columns do not change.

6.

No. The first column does not change.

7.

No. The third and the fourth columns do not change.

8.

No. The third and the fourth columns do not change.