Name: The Eigenvectors and Eigenvalues
Uploaded: 2019-07-19
Duration: 17 min 15 s
Description: Studying linear algebra, we will inevitably stumble upon the concept of eigenvalues. These sound very exotic, but they are very important.

Question

Given the matrix

A = [\begin{matrix} - 1 & 3 & 5 \\ 1 & - 3 & - 5 \\ - 1 & 3 & 5 \end{matrix}]

and

X

be the solution set of the equation

A^{x} = A,

where

x \in N - \{1\}

. Evaluate

\prod \frac{x^{3} + 1}{x^{3} - 1},

where the continued product extends

\forall x \in X

.

Accepted Answer

It is given that $A = [\begin{matrix} - 1 & 3 & 5 \\ 1 & - 3 & - 5 \\ - 1 & 3 & 5 \end{matrix}]$ .

Now, $A^{2} = [\begin{matrix} - 1 & 3 & 5 \\ 1 & - 3 & - 5 \\ - 1 & 3 & 5 \end{matrix}] [\begin{matrix} - 1 & 3 & 5 \\ 1 & - 3 & - 5 \\ - 1 & 3 & 5 \end{matrix}] = [\begin{matrix} - 1 & 3 & 5 \\ 1 & - 3 & - 5 \\ - 1 & 3 & 5 \end{matrix}] = A$ .

Similarly, $A^{3} = A^{4} = . . . . . . = A^{x} = A \forall x \in N - \{1\}$ .

Now, it is also given that $A^{x} = A$ for $x = \{2, 3, 4, . . . . . . . . . . ., \infty\}$ .

$∴ \prod_{x = 2}^{\infty} \frac{x^{3} + 1}{x^{3} - 1}$

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

$= \prod_{x = 2}^{\infty} \{\frac{(x + 1)}{(x - 1)} \times \frac{(x^{2} - x + 1)}{(x^{2} + x + 1)}\}$

$= \lim_{n \to \infty} \{(\frac{3}{1} \times \frac{4}{2} \times \frac{5}{3} \times \frac{6}{4} \times \frac{7}{5} \times \frac{8}{6} \times . . . . . . \times \frac{n}{n - 2} \times \frac{n + 1}{n - 1}) (\frac{3}{7} \times \frac{7}{13} \times \frac{13}{21} \times . . . . . . \times \frac{n^{2} - n + 1}{n^{2} + n + 1})\}$

$= \lim_{n \to \infty} \{(\frac{n (n + 1)}{2}) (\frac{3}{n^{2} + n + 1})\}$

$= \lim_{n \to \infty} \frac{3 (n^{2} + n)}{2 (n^{2} + n + 1)}$

$= \frac{3}{2} \times \lim_{n \to \infty} \frac{(\frac{n^{2} + n}{n^{2}})}{(\frac{n^{2} + n + 1}{n^{2}})}$

$= \frac{3}{2} \times \lim_{n \to \infty} \frac{(1 + \frac{1}{n})}{(1 + \frac{1}{n} + \frac{1}{n^{2}})}$

$= \frac{3}{2} \times \frac{(1 + 0)}{(1 + 0 + 0)}$

$= \frac{3}{2}$

Given the matrix $A = [\begin{matrix} - 1 & 3 & 5 \\ 1 & - 3 & - 5 \\ - 1 & 3 & 5 \end{matrix}]$ and $X$ be the solution set of the equation $A^{x} = A,$ where $x \in N - \{1\}$ . Evaluate $\prod \frac{x^{3} + 1}{x^{3} - 1},$ where the continued product extends $\forall x \in X$ .

Important Questions on Matrices and Determinants

Learn and Prepare for any exam you want !

Given the matrix A=-1351-3-5-135 and X be the solution set of the equation Ax=A, where x∈N-1. Evaluate ∏x3+1x3-1, where the continued product extends ∀x∈X.

Important Questions on Matrices and Determinants

Learn and Prepare for any exam you want !

Given the matrix $A = [\begin{matrix} - 1 & 3 & 5 \\ 1 & - 3 & - 5 \\ - 1 & 3 & 5 \end{matrix}]$ and $X$ be the solution set of the equation $A^{x} = A,$ where $x \in N - \{1\}$ . Evaluate $\prod \frac{x^{3} + 1}{x^{3} - 1},$ where the continued product extends $\forall x \in X$ .