It is currently Sun Mar 18, 2018 6:17 pm

 All times are UTC [ DST ]

 Page 1 of 1 [ 2 posts ]
 Print view Previous topic | Next topic
Author Message
 Post subject: Determinant of a matrixPosted: Sat Jun 25, 2016 7:53 am

Joined: Sat Nov 07, 2015 6:12 pm
Posts: 828
Location: Larisa
Let $A$ be an $n \times n$ matrix that is defined as: $$A=a_{ij}=\left\{\begin{matrix} 5\,, & i=j \\ 2\,, & i<j\\ -2\,, &i>j \end{matrix}\right.$$ If $D_n$ is its determinant then prove that $D_n=10D_{n-1}-21D_{n-2}$ and in continunation evaluate the det of the matrix.

_________________
Imagination is much more important than knowledge.

Top

 Post subject: Re: Determinant of a matrixPosted: Sat Jun 25, 2016 7:55 am
 Team Member

Joined: Mon Nov 09, 2015 1:52 pm
Posts: 426
$\displaystyle{D_{1}=\left|5\right|=5}$

$\displaystyle{D_{2}=\begin{vmatrix} 5 & 2\\ -2 & 5 \end{vmatrix}=25+4=29}$

Let $\displaystyle{n\geq 3}$ .

If $\displaystyle{\Gamma_{i}\,,\Sigma_{i}\,,1\leq i\leq n}$ are the lines and the columns, repsectively, of the matrix, then

by using the operation $\displaystyle{\Sigma_{1}\to \Sigma_{1}+\Sigma_{n}}$ we get :

$\displaystyle{D_{n}=\begin{vmatrix} 5 & 2 & 2 & ... &2 &2 \\ -2 &5 &2 &... & 2 &2 \\ -2 &-2 &5 &... &2 &2 \\ -2 &-2 &-2 &5 &... &2 \\ ... &... &... &... &... &... \\ -2 &-2 &-2 &... &-2 &5 \end{vmatrix}=\begin{vmatrix} 7 & 2 & 2 & ... &2 &2 \\ 0 &5 &2 &... & 2 &2 \\ 0 &-2 &5 &... &2 &2 \\ 0 &-2 &-2 &5 &... &2 \\ ... &... &... &... &... &... \\ 3 &-2 &-2 &... &-2 &5 \end{vmatrix}}$

and then, by using $\displaystyle{\Gamma_{1}\to \Gamma_{1}+\Gamma_{n}}$, we have that :

\displaystyle{\begin{aligned} D_{n}&=\begin{vmatrix} 10 & 0 & 0 & ... &0 &7 \\ 0 &5 &2 &... & 2 &2 \\ 0 &-2 &5 &... &2 &2 \\ 0 &-2 &-2 &5 &... &2 \\ ... &... &... &... &... &... \\ 3 &-2 &-2 &... &-2 &5 \end{vmatrix}\\&=10\cdot \begin{vmatrix} 5 &2 &... & 2 &2 \\ -2 &5 &... &2 &2 \\ -2 &-2 &5 &... &2 \\ ... &... &... &... &... \\ -2 &-2 &... &-2 &5 \end{vmatrix}+3\,(-1)^{n+1}\cdot \begin{vmatrix} 0 & 0 & ... &0 &7 \\ 5 &2 &... & 2 &2 \\ -2 &5 &... &2 &2 \\ -2 &-2 &5 &... &2 \\ ... &... &... &... &... \\ -2 &-2 &... &-2 &5 \end{vmatrix}\\&=10\,D_{n-1}+3\,(-1)^{n+1}\,7\,(-1)^{1+n-1}\cdot \begin{vmatrix} 5 &2 &... & 2 \\ -2 &5 &... &2 \\ -2 &-2 &5 &... \\ ... &... &... &... \\ -2 &-2 &... &5 \end{vmatrix}\\&=10\,D_{n-1}-21\,D_{n-2}\end{aligned}}

We observe that $\displaystyle{D_{1}=5=\dfrac{7+3}{2}\,,D_{2}=29=\dfrac{7^2+3^2}{2}}$ .

Suppose that $\displaystyle{D_{k}=\dfrac{7^{k}+3^{k}}{2}\,,\forall\,k\in\left\{1,...,n-1\right\}\,(I)}$ .

$\displaystyle{\bullet\,k=n}$

\displaystyle{\begin{aligned} D_{n}&=10\,D_{n-1}-21\,D_{n-2}\\&\stackrel{(I)}{=}10\cdot \dfrac{7^{n-1}+3^{n-1}}{2}-21\cdot \dfrac{7^{n-2}+3^{n-2}}{2}\\&=5\,7^{n-1}+5\,3^{n-1}-\dfrac{21}{2}\,7^{n-2}-\dfrac{21}{2}\,3^{n-2}\\&=7^{n-2}\,\left[5\cdot 7-\dfrac{21}{2}\right]+3^{n-2}\,\left[5\cdot 3-\dfrac{21}{2}\right]\\&=\left(35-\dfrac{21}{2}\right)\,7^{n-2}+\left(15-\dfrac{21}{2}\right)\,3^{n-2}\\&=\dfrac{49}{2}\,7^{n-2}+\dfrac{9}{2}\,3^{n-2}\\&=\dfrac{1}{2}\,\left(7^2\cdot 7^{n-2}+3^2\cdot 3^{n-2}\right)\\&=\dfrac{7^{n}+3^{n}}{2}\end{aligned}} .

So, by induction, $\displaystyle{D_{n}=\dfrac{7^{n}+3^{n}}{2}\,,n\in\mathbb{N}}$ .

Top

 Display posts from previous: All posts1 day7 days2 weeks1 month3 months6 months1 year Sort by AuthorPost timeSubject AscendingDescending
 Page 1 of 1 [ 2 posts ]

 All times are UTC [ DST ]

#### Mathimatikoi Online

Users browsing this forum: No registered users and 1 guest

 You cannot post new topics in this forumYou cannot reply to topics in this forumYou cannot edit your posts in this forumYou cannot delete your posts in this forumYou cannot post attachments in this forum

Search for:
 Jump to:  Select a forum ------------------ Algebra    Linear Algebra    Algebraic Structures    Homological Algebra Analysis    Real Analysis    Complex Analysis    Calculus    Multivariate Calculus    Functional Analysis    Measure and Integration Theory Geometry    Euclidean Geometry    Analytic Geometry    Projective Geometry, Solid Geometry    Differential Geometry Topology    General Topology    Algebraic Topology Category theory Algebraic Geometry Number theory Differential Equations    ODE    PDE Probability & Statistics Combinatorics General Mathematics Foundation Competitions Archives LaTeX    LaTeX & Mathjax    LaTeX code testings Meta