# Matrices and determinants, by applying properties to find the value of the following determinant.

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The properties that we will use are: The determinant of the transposed is equal to: 
\begin{align}&|B^t|=|B|\end{align}is The determinant of the inverse is the inverse of the determinant:


\begin{align}&|B^{-1}| = |B|^{-1}\end{align}is The determinant of the product is the product of the determinants:


\begin{align}&|(A\cdot B| = |A|\cdot |B|\end{align}The determinant of a matrix by a scalar is the determinant of the matrix multiplied by the scalar raised to the dimension of the array. If the matrix B is square, of dimension 3, then


\begin{align}&|2\cdot B|=2^3\cdot|B|\end{align}here We go:

The first sum is


\begin{align}&|2B^{-1}A|=|2B^-1||A|=\\&=2^3|B^{-1}||A|=8·|B|·|A|=\\&=8·(-1)·5=-40\\&\end{align}The second summand is


\begin{align}&|(A^2B^t|=|A^2||B^t|=\\&=|A^2||B|=|A||A||B|=\\&=5^2·(-1)=-25\end{align}, therefore, the result is


\begin{align}&-40-(-25)=-40+25=-15\end{align}property you can check in [Determinant, rank-and-under (theory)](https://www.matesfacil.com/matrices/determinante-rango.html).

A greeting to you, friend. I leave some links of problems of matrix algebra:

- [Product of matrices](https://www.matesfacil.com/matrices/resueltos-matrices-producto.html)
- [Powers of arrays](https://www.matesfacil.com/matrices/resueltos-matrices-potencias.html)
- [Determinants of matrices](https://www.matesfacil.com/matrices/resueltos-matrices-determinantes.html)
- [Index of matrix algebra](https://www.matesfacil.com/matrices.htm)