What is the value of the determinant ?

Engineering

Mathematics

Introduction to Determinants

Properties of Determinant

Question

What is the value of the determinant $|\begin{array}{ccc} 1! & 2! & 3! \\ 2! & 3! & 4! \\ 3! & 4! & 5! \end{array}|$ ?

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Given,

|\begin{array}{ccc} 1! & 2! & 3! \\ 2! & 3! & 4! \\ 3! & 4! & 5! \end{array}|

= 1! \cdot \det (\begin{array}{cc} 3! & 4! \\ 4! & 5! \end{array}) - 2! \cdot \det (\begin{array}{cc} 2! & 4! \\ 3! & 5! \end{array}) + 3! \cdot \det (\begin{array}{cc} 2! & 3! \\ 3! & 4! \end{array})

= 11 · 144 – 2! · 96 + 3! · 12

= 144 – 192 + 72

= 24