Matrices & Determinants Question - None

Question

If Δ(x) =

|\begin{matrix} 0 & 2 x - 2 & 2 x + 8 \\ x - 1 & 4 & x^{2} + 7 \\ 0 & 0 & x + 4 \end{matrix})

and f(x) =

\sum_{i = 1}^{3} \sum_{j = 1}^{3} a_{i j} c_{i j}

, where a_{ij} is the element of i^th row and j^th column in Δ(x) and c_{ij} is the cofactor of a_{ij} ∀ i and j, then find the greatest value of f(x), where x ∈ [-3, 18].

Accepted Answer

The expression f(x) = \Sigma_{i=1}^3 \Sigma_{j=1}^3 a_{ij} c_{ij} represents 3 times the determinant of the matrix \Delta(x). Calculating the determinant \Delta(x) by expanding along the first column: \Delta(x) = -(x-1) * [(2x-2)(x+4) - 0] = -(x-1)(2x-2)(x+4) = -2(x-1)^2(x+4). Then f(x) = 3 * \Delta(x) = -6(x-1)^2(x+4). For x \in  [-3, 18], we analyze the function f(x). Since (x-1)^2 \geq 0 and (x+4) \geq 1 for x \in  [-3, 18], f(x) \leq 0. The maximum value is 0, which occurs at x = 1.

Question:

Step-by-Step Solution

Master Matrices & Determinants with Mathbee