- #1
mathmari
Gold Member
MHB
- 5,049
- 7
Hey!
I want to calculate the determinant of the following matrices:
I have done the following:
Could you give me a hint how we could calculate the determinant of $B$ and $C$ ? (Wondering)
I want to calculate the determinant of the following matrices:
- $$A=\begin{pmatrix}-3 & -11 & -11 & 45 \\ 1 & 11 & 10 & -83 \\ 1 & -6 & -5 & 81 \\ 0 & -3 & -3 & 42\end{pmatrix}$$
- $$B=\begin{pmatrix}1+a_1 & a_2 & \ldots & a_n \\ a_1 & 1+a_2 & \ldots & a_n \\ \ldots & \ldots & \ldots & \ldots \\ a_1 & a_2 & \ldots & 1+a_n\end{pmatrix}$$
- $$C=(c_{ij})\in \mathbb{Q}^{n\times n} \text{ with } c_{ij}=\left\{\begin{matrix}
0 & i=j\\
1 & i\neq j
\end{matrix}\right.$$
I have done the following:
- $$|A|=\begin{vmatrix}-3 & -11 & -11 & 45 \\ 1 & 11 & 10 & -83 \\ 1 & -6 & -5 & 81 \\ 0 & -3 & -3 & 42\end{vmatrix}=-3\begin{vmatrix} 11 & 10 & -83 \\ -6 & -5 & 81 \\ -3 & -3 & 42\end{vmatrix}-\begin{vmatrix} -11 & -11 & 45 \\ -6 & -5 & 81 \\ -3 & -3 & 42\end{vmatrix}+\begin{vmatrix} -11 & -11 & 45 \\ 11 & 10 & -83 \\ -3 & -3 & 42\end{vmatrix}-0\cdot \begin{vmatrix} -11 & -11 & 45 \\ 11 & 10 & -83 \\ -6 & -5 & 81 \end{vmatrix} =\ldots =42$$
Could you give me a hint how we could calculate the determinant of $B$ and $C$ ? (Wondering)