Determinant of 5x5 matrix with letters

Question

I want to find the determinant of the following $5 \times 5$ matrix

\begin{bmatrix} 2 & 0 & p & 0 & q \\r & 2 & s & 1 & 2 \\0 & 0 & 1 & 0 & 0\\u & 1 & v & 1 & w\\ 0 & 0 & x & 0 & 2 \\\end{bmatrix}

I know I have to do some row/column operations and expansions but I really don't get it. Help is appreciated. Thanks in advance :)

Answer 1

Let given matrix is A.

\begin{bmatrix} 2 & 0 & p & 0 & q \\r & 2 & s & 1 & 2 \\0 & 0 & 1 & 0 & 0\\u & 1 & v & 1 & w\\ 0 & 0 & x & 0 & 2 \\\end{bmatrix}

Expand it by $A_{33}$

= 1 × \begin{bmatrix} 2 & 0 & 0 & q \\r & 2 & 1 & 2 \\u & 1 & 1 & w\\ 0 & 0 & 0 & 2 \\\end{bmatrix}

Expand it by $A_{44}$

= 1 × 2 × \begin{bmatrix} 2 & 0 & 0 \\r & 2 & 1 \\u & 1 & 1 \\\end{bmatrix}

Expand it by $A_{11}$

= 1 × 2 × (2 - 1)

= 2