Introduction to Determinants

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Question

If A is a square matrix of order 2 x 2 such that |A| = 27, then sum of the infinite series $| A | + |\frac{1}{2} A| + |\frac{1}{4} A| + |\frac{1}{8} A| + \dots$ is equal to __________.

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Solution

$\begin{array}{l} | A | + |\frac{1}{2} A| + |\frac{1}{4} A| + |\frac{1}{8} A| + \dots \\ \Rightarrow | A | + \frac{1}{4} | A | + \frac{1}{16} | A | + \dots \\ \Rightarrow \frac{| A |}{1 - \frac{1}{4}} = \frac{4}{3} | A | = 36 \end{array}$

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