Expand a determinant

Question

I want to expand the determinant on the left side of the following equation: $$ \left| \begin{array}{ccc} 1 & 0 & 0 \\ p & q & -1 \\ \text{a}-x & -y & -z \end{array} \right|=0.$$

I am getting $qz+y=0.$ Please verify if this is correct.

Answer 1

You may use Wolfram Alpha to check this kind of computations.

Answer 2

Expanding the determinant about the first row, we obtain: $$\begin{vmatrix} 1 & 0 & 0 \\ p & q & -1 \\ a-x & -y & -z \end{vmatrix} = \begin{vmatrix} q & -1 \\ -y & -z \end{vmatrix} = q(-z) -(-1)(-y) = -(qz+y)$$

Answer 3

The determinant of a matrix can be written by a cofactor expansion across a specific row or column (any row of column for that matter).

In your case, the determinant can be chosen to be expanded on the first row (which is probably the easiest), yielding:

$$\begin{vmatrix}1 & 0 & 0 \\ p & q & -1 \\a-x & -y & -z\end{vmatrix} = -qz - y = -(qz + y)$$

Determinant expansion