For what real number r is the matrix nonsingular?

Question

$$A = \begin{bmatrix} 1 & r & 0 & 0 \\ r & 1 & 0 & 0 \\ 0 & r & 1 & 0 \\ 0 & 0 & r & 1 \end{bmatrix} $$ For what real number $r$ is the matrix nonsingular?

Good day! I am new here and am having trouble how to approach this problem. Should I just simply find the determinant and from there gain equations that would allow me to solve for $r$ in the matrix?

Or is there a simpler or easier way to go about it? Any help is much appreciated. Thank you.

Answer 1

The easiest way is, imo, to do $\;C_2-rC_1\;$ , with $\;C_i=\;$ the i-th column. then your matrix becomes

$$\begin{pmatrix} 1 & 0 & 0 & 0 \\ r & 1-r^2 & 0 & 0 \\ 0 & r & 1 & 0 \\ 0 & 0 & r & 1 \end{pmatrix}$$

and (1) since elementary operations that only involve adding a multple of some column (row) to another column (row) do not change the matrix's determinant, and (2) the determinant of a triangular matrix is just the product of its elements on the main diagonal, we can see at once that your matrix's determinant is $\;1-r^2\;$ ...

Answer 2

Computing the determinant is easy for this one, you can save time by using the fact that the determinant of a triangular matrix is the product of its diagonals.

Answer 3

We can find the determinant easily by Laplace expansion, i.e., $$ \det(A)=(r+1)(r-1). $$