Question

Let${\mathit{M}}_{\mathbf{n}}$ be therole="math" localid="1660405225698" $\mathit{n}\mathbf{\times }\mathit{n}$ matrix with l's on the main diagonal and directly above the main diagonal,role="math" localid="1660405220979" $\mathbf{-}\mathbf{1}$'s directly below the main diagonal, androle="math" localid="1660405229621" $\mathbf{0}$'s elsewhere. For example ${\mathit{M}}_{\mathbf{4}}\mathbf{=}\left[\begin{array}{cccc}1& 1& 0& 0\\ -1& 1& 1& 0\\ 0& -1& 1& 1\\ 0& 0& -1& 1\end{array}\right]$,

Let ${\mathit{d}}_{\mathbf{n}}\mathbf{=}\mathit{d}\mathit{e}\mathit{t}\left(M_n\right)$

a. For $\mathit{n}\mathbf{⩾}\mathbf{3}$, find a formula expressing ${\mathit{d}}_{\mathbf{n}}$ in terms of${\mathit{d}}_{\mathbf{n}\mathbf{-}\mathbf{1}}$ and${\mathit{d}}_{\mathbf{n}\mathbf{-}\mathbf{2}}$.
b. Find ${\mathit{d}}_{\mathbf{1}}\mathbf{,}{\mathit{d}}_{\mathbf{2}}\mathbf{,}{\mathit{d}}_{\mathbf{3}}\mathbf{,}{\mathit{d}}_{\mathbf{4}}$, and ${\mathit{d}}_{\mathbf{10}}$.
c. For which positive integers$\mathit{n}$ is the matrix${\mathit{M}}_{\mathbf{n}}$ invertible?

Short answer

Therefore,

a) $d_n=d_{n-1}+d_{n-2}.$

b) $d_1=1,d_2=2,d_3=3,d_4=5,d_{10}=89$

c) $M_n$is invertible for all $n\in \mathrm{\mathbb{N}}$.

Step-by-step solution

Step by Step Solution: Step 1: Matrix Definition

Matrix is aset of numbers arranged in rowsand columns so as to form a rectangular array.

The numbers are called the elements, or entries, of the matrix.

If there are $m$rows and $n$columns, the matrix is said to be a “ $m$by $n$” matrix, written “ $m\times n$.”

(a)Step 2: To find the formula expressing in dn terms

Using the Laplace expansion along the first column, we have:

$$\begin{gathered} d_n=1d_{n-1}-(-1)·1·d_{n-2} \\ {d}_n=d_{n-1}+d_{n-2}. \end{gathered}$$

(b)Step 3: To find d1,d2,d3,d4 and d10.

It applies that:

$$\begin{gathered} d_n=d_{n-1}+d_{n-2}. \\ {d}_1=1, \\ {d}_2=2, \\ {d}_3=3, \\ {d}_4=5, \\ {d}_{10}=89. \end{gathered}$$

(c)Step 4: To find which positive integers matrix Mn of is invertible

Given the formula for recursion $d_n=d_{n-1}+d_{n-2}$, $M_n$is invertible for all $n\in \mathrm{\mathbb{N}}$ .