Question

Let I denote the $3 \times 3$ identity matrix and P be a matrix obtained by rearranging the columns of I. Then

• A. There are six distinct choices for P and det (P) = 1
• B. There are six distinct choices for P and det (P) = $\pm$ 1
• C. There are more than one choices for P and some of them are invertible
• D. There are more than one choices for P and $P^1$ = I is each choice

Answer 1

Answer: (B)

There are six distinct choices for P and det (P) = $\pm$ 1

Answer 2

$I_{3 \times3} = \begin{bmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\end{bmatrix}$ 3 different columns can be arranged in 3! i.e., 6 ways. In each case, if there are even number of interchanges of columns, determinant remains 1 and for odd number of interchange determinant takes the negative value i.e., -1.