Answer
$abcde$
Work Step by Step
Step 1. Start from the original, use the first row to expand and get: $\begin{array}( \\A=\begin{vmatrix} a&0&0&0&0\\0&b&0&0&0\\0&0&c&0&0\\0&0&0&d&0\\0&0&0&0&e \end{vmatrix}=a \\ \\ \end{array} \begin{vmatrix} b&0&0&0\\0&c&0&0\\0&0&d&0\\0&0&0&e \end{vmatrix}$
Step 2. Again use the first row to expand and get: $\begin{array}( \\A=ab \\ \\ \end{array} \begin{vmatrix}c&0&0\\0&d&0\\0&0&e \end{vmatrix}$
Step 3. Repeat the above steps to get: $\begin{array}( \\A=abc \\ \\ \end{array} \begin{vmatrix}d&0\\0&e \end{vmatrix}$
Step 4. Calculate the last determinant to get: $A=abcde$
