First slide

Applications of determinant

Question

Let a and b be two real numbers such that $a > 1, b > 1.$ $I f A = (\begin{matrix} a & 0 \\ 0 & b \end{matrix})$ then $\lim_{n \to \infty} A^{- n}$ is

Moderate

A

unit matrix
B

null matrix
C

$2 I$
D

none of these

Solution

$\begin{array}{l} A = (\begin{array}{ll} a & 0 \\ 0 & b \end{array}) \\ \begin{aligned} \Rightarrow A^{2} & = (\begin{array}{ll} a & 0 \\ 0 & b \end{array}) (\begin{array}{ll} a & 0 \\ 0 & b \end{array}) \\ = (\begin{array}{ll} a^{2} & 0 \\ 0 & b^{2} \end{array}) \\ \Rightarrow A^{3} & = (\begin{array}{cc} a^{2} & 0 \\ 0 & b^{2} \end{array}) (\begin{array}{ll} a & 0 \\ 0 & b \end{array}) \\ = (\begin{array}{cc} a^{3} & 0 \\ 0 & b^{3} \end{array}) \end{aligned} \end{array}$
$\begin{array}{l} \Rightarrow A^{n} = (\begin{array}{cc} a^{n} & 0 \\ 0 & b^{n} \end{array}) \\ \Rightarrow {(A^{n})}^{- 1} = \frac{1}{a^{n} b^{n}} (\begin{array}{cc} b^{n} & 0 \\ 0 & a^{n} \end{array}) = (\begin{array}{cc} a^{- n} & 0 \\ 0 & b^{- n} \end{array}) \\ \Rightarrow lim_{n \to \infty} {(A^{n})}^{- 1} = (\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}) as a > 1 and b > 1 \end{array}$

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