First slide

Introduction to Determinants

Question

If $f (x) = |\begin{array}{ccc} x - 3 & 2 x^{2} - 18 & 3 x^{3} - 81 \\ x - 5 & 2 x^{2} - 50 & 4 x^{3} - 500 \\ 1 & 2 & 3 \end{array}|$ ,then $f (1) \cdot f (3) + f (3) \cdot f (5) \cdot f (5) \cdot f (1)$ is equal to

Moderate

A

f(1)
B

f(3)
C

f(1)+f(3)
D

f(1)+f(5)

Solution

$\begin{array}{l} f (1) = |\begin{array}{rrr} - 2 & - 16 & - 78 \\ - 4 & - 48 & - 496 \\ 1 & 2 & 3 \end{array}| \\ f (3) = |\begin{array}{rrr} 0 & 0 & 0 \\ - 2 & - 32 & - 392 \\ 1 & 2 & 3 \end{array}| = 0 \end{array}$
$\begin{array}{c} f (1) f (3) + f (3) f (5) f (5) f (1) = f (1) f (3) [1 + {f (5)}^{2}] \\ = 0 \times [1 + {f (5)}^{2}] \\ = 0 \end{array}$

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