How to Calculate Determinants for a Matrix with Linear Algebra?

[Perrry]

Let $$\begin{gather*}A_n\end{gather*}$$ be an nxn matrix with the matrixelement $$\begin{gather*}a_ik\end{gather*}$$=i+k, i, k = 1, ... ,n. Decide for every value the n-determinant $$\begin{gather*}D_n\end{gather*}$$ = det($$\begin{gather*}A_n\end{gather*}$$). Don´t forget the value of n=1.

We are two guys here at home that don´t get it right. What shall we start with? We are both newbies on this!

Thanks in advance

Perrry

 

[radou]

There is a place called 'Linear & Abstract Algebra' for such threads.

 

[kyle8921]

Dear Perrry,

Calculating determinants for a matrix is an important concept in linear algebra. To start, we need to understand the basic definition of a determinant. A determinant is a numerical value that can be calculated from a square matrix. In simple terms, it represents the scaling factor of the matrix.

To calculate the determinant of an n x n matrix, we can use the following formula:

\begin{gather*}D_n = \sum_{j=1}^{n} (-1)^{i+j} a_{ij}M_{ij}\end{gather*}

Where \begin{gather*}a_{ij}\end{gather*} is the element in the i-th row and j-th column, and \begin{gather*}M_{ij}\end{gather*} is the minor of \begin{gather*}a_{ij}\end{gather*}, which is the determinant of the submatrix obtained by removing the i-th row and j-th column.

In the given matrix \begin{gather*}A_n\end{gather*}, we can see that \begin{gather*}a_{ik} = i+k\end{gather*}. So, we can rewrite the formula as:

\begin{gather*}D_n = \sum_{j=1}^{n} (-1)^{i+j} (i+k)M_{ij}\end{gather*}

Now, let's calculate the determinant for n = 1. In this case, the matrix \begin{gather*}A_1\end{gather*} will have only one element, which is \begin{gather*}a_{11} = 1+1 = 2\end{gather*}. Using the formula, we get:

\begin{gather*}D_1 = (-1)^{1+1} (2)M_{11} = 2M_{11}\end{gather*}

Since \begin{gather*}M_{11}\end{gather*} is the determinant of a submatrix with no elements, it is equal to 1. Therefore, \begin{gather*}D_1 = 2\end{gather*}.

For n = 2, the matrix \begin{gather*}A_2\end{gather*} will look like:

\begin{gather*}A_2 = \begin{pmatrix}