Let A = \left[\right. x^{2} + 1 & x + 5 & 5 x + 7 \\ y + 3 & 2 - x & z + 4 \\ 3 & 5 & 1 - x \left]\right. \textrm{ }\textrm{ } and \textrm{ }\textrm{ } \left|\right. A \left|\right. = 2 . If B = adj A and f(x) = tr (B–1) then find global maximum value of f(x) in [0, 2].

Engineering

Mathematics

Inverse of a Matrix

Maxima and Minima

Question

Let $A = [\begin{matrix} x^{2} + 1 & x + 5 & 5 x + 7 \\ y + 3 & 2 - x & z + 4 \\ 3 & 5 & 1 - x \end{matrix}] and | A | = 2 .$

If B = adj A and f(x) = tr (B^–1) then find global maximum value of f(x) in [0, 2].

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$∵ B = adjA = | A | (\frac{1}{| A |} adj A)$

⇒ B = |A| A^–1 ⇒ B = 2A^–1

$\Rightarrow B^{- 1} = \frac{1}{2} A$

$∴$ f(x) = tr (B^–1) = $\frac{1}{2} tr (A)$

$= \frac{1}{2} (x^{2} - 2 x + 4)$

$f (x) = \frac{1}{2} ({(x - 1)}^{2} + 3)$

$∴$ Global max = f(0) = f(2) = 4/2 = 2