# How do I read this matrix and find the determinant?

Here is the matrix, which I'm asked to give the determinant of. How do I read this?

$$R^{(1)}:= \begin{bmatrix} 1 \\ & \ddots \\ & & 0 & \cdots & 1 & &\\ & & \vdots & \ddots & \vdots \\ & & 1 & \cdots & 0\\ & & & & & \ddots \\ & & & & & & 1 \end{bmatrix}$$

How do I deal with the dots and find the determinant?

• @FraGrechi I'm not given the dimensions, but the problem does mention row operations on a quadratic matrix; however, I'm not sure how that fits with this question. – cdignam Mar 21 '17 at 1:39
• I have never heard of the term "quadratic matrix", what does this refer to in the course/book you are currently taking/reading? Does this mean that you are dealing with an $n \times n$ square matrix? – FraGrechi Mar 21 '17 at 1:53
• @FraGrechi Yeah, it means square matrix. – cdignam Mar 21 '17 at 1:54
• Oh I am an idiot, of course it is a square matrix. To find the determinant simply swap those two rows, obtain $I$, and find that $\det R = -1$. – FraGrechi Mar 21 '17 at 2:02
• @FraGrechi Oh, thanks. That's what I needed. – cdignam Mar 21 '17 at 2:06

As FraGrechi said, we can transform this matrix into the identity matrix, whose determinant is $1$; however, since we swap two rows, we need to multiply this by $-1$, giving us our final determinate of $-1$.