<s>[INST] What is a determinant in matrix ? NOT how to get it but WHAT is it in real life We know to get determinant in a matrix we use sarrus, laplace, x methods and so on
But question : WHAT is it, according to some pages is about permutations the products of element of the matrix , but WHAT IS IT ?
 [/INST] \begin{align}
& \left[ \begin{array}{c} x \\ y \end{array} \right] \mapsto \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right]\left[ \begin{array}{c} x \\ y \end{array} \right] = \left[ \begin{array}{c} ax+by \\ cx+dy \end{array} \right] \\[8pt]
& \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] \mapsto \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right]\left[ \begin{array}{c} 1 \\ 0 \end{array} \right] = \left[ \begin{array}{c} a \\ c \end{array} \right] \\[8pt]
& \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] \mapsto \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right]\left[ \begin{array}{c} 0 \\ 1 \end{array} \right] = \left[ \begin{array}{c} b \\ d \end{array} \right]
\end{align}
Draw the arrows representing $\left[ \begin{array}{c} 1 \\ 0 \end{array} \right]$ and $\left[ \begin{array}{c} 0 \\ 1 \end{array} \right].$ From the head of the arrow $\left[ \begin{array}{c} 1 \\ 0 \end{array} \right],$ draw another arrow parallel to $\left[ \begin{array}{c} 0 \\ 1 \end{array} \right]$ and from the head of the arrow $\left[ \begin{array}{c} 0 \\ 1 \end{array} \right]$ draw another parallel to $\left[ \begin{array}{c} 1 \\ 0 \end{array} \right].$ There you have a square whose area is $1.$
Then do the same with $\left[ \begin{array}{c} a \\ c \end{array} \right]$ and $\left[ \begin{array}{c} b \\ d \end{array} \right]$ and get a parallelogram. The area of that parallelogram is $\left|\det\left[ \begin{array}{cc} a & b \\ c & d \end{array} \right]\right|.$ The determinant is positive if turning from $\left[ \begin{array}{c} a \\ c \end{array} \right]$ to $\left[ \begin{array}{c} b \\ d \end{array} \right]$ amounts to turning counterclockwise, and negative if clockwise. In other words, positive if the parallelogram has the same orientation as the square, and negative if the orientation is reversed.
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