Easy Determinant(s) !

Algebra Level 2

∣ a h + b g g c − a 2 a b + c h b f + b a f c − a h b h + b c a f + b c c 2 − a g b g + f c ∣ = ? \left | \begin{array}{ccc} ah+bg & gc-a^2 & ab+ch \\ bf+ba & fc-ah & bh+bc \\ af+bc & c^2-ag & bg+fc \\ \end{array} \right | = ?

a b c f g h abcfgh 0 0 a b c abc a b c + f g h abc+fgh

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1 solution

The given determinant can be written as a product of two determinants :

∣ h g a b f h f c g ∣ ∣ a b 0 0 c − a c 0 b ∣ \left | \begin{array}{ccc} h & g & a \\ b & f & h \\ f & c & g \\ \end{array} \right | \left | \begin{array}{ccc} a & b & 0 \\ 0 & c & -a\\ c & 0 & b \\ \end{array} \right |

The second determinant is 0 0 and hence the product is 0 0 .

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