Question

Find the determinant of $\left[\begin{array}{ccc}a&b&c\\b&c&a\\c&a&b\end{array}\right]$

Answer 1

Answer:

$\text {Determinant = } 3abc - a^3 - b^3 -c^3$

Step-by-step explanation:

$\text{Find the determinant of} \left[\begin{array}{ccc}a&b&c\\b&c&a\\c&a&b\end{array}\right]$

$\text {Determinant = } a \times \left|\begin{array}{ccc}c&a\\a&b\end{array}\right| - b \times \left|\begin{array}{ccc}b&a\\c&b\end{array}\right| + c \times \left|\begin{array}{ccc}b&c\\a&b\end{array}\right|$

$\text {Determinant = } a(bc - a^2) - b(b^2 - ac) + c(ab -c^2)$

$\text {Determinant = } abc - a^3 - b^3 + abc + abc -c^3$

$\text {Determinant = } 3abc - a^3 - b^3 -c^3$