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Introduction to Determinants

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Question

let the numbers 2, b, c are in A.P and $A = [\begin{matrix} 1 & 1 & 1 \\ 2 & b & c \\ 4 & b^{2} & c^{2} \end{matrix}] .$ If det $(A) \in [2, 16]$ then the maximum value of c is

Moderate

Solution

We have
$\begin{array}{l} det A = (b - 2) (c - 2) (c - b) where b = \frac{c + 2}{2} (∵ |\begin{matrix} 1 & 1 & 1 \\ a & b & c \\ a^{2} & b^{2} & c^{2} \end{matrix}| = (a - b) (b - c) (c - a)) \\ = (\frac{c + 2}{2} - 2) (c - 2) (c - (\frac{c + 2}{2})) \\ = \frac{(c - 2)^{3}}{4} \\ G i v e n 2 \leq d e t A \leq 16 \\ \Rightarrow 2 \leq \frac{(c - 2)^{3}}{4} \leq 16 \\ \Rightarrow 8 \leq (c - 2)^{3} \leq 64 \\ \Rightarrow 2 \leq c - 2 \leq 4 \\ \Rightarrow 4 \leq c \leq 6 . \end{array}$

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If [ ] denotes the greatest integer less than or equal to the real number under consideration, and $- 1 \leq x < 0, 0 \leq y < 1$ , $1 \leq z < 2$ , then the value of the determinant $|\begin{array}{ccc} [x] + 1 & [y] & [z] \\ [x] & [y] + 1 & [z] \\ [x] & [y] & [z] + 1 \end{array}|$ is

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