I have completed by Matrices classes and the only question that is roaming in my mind is what is this determinant actually. Like they simply told the formula and things that are to be calculated and how to calculate them.

I wanna know what these determinants actually tell about a matrix and how do they relate to them? Like we have a matrix and its determinant, how is this determinant related to it? And why do we calculate the determinant in the way we calculate it?

Like already matrix multiplication is a bit different then I expected it to be. One more thing I wanna know is why the determinant has such properties and how does it exhibit them? And how can these things be related to each other? Thanks!

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1 Answers1


An $n\times n$ matrix with real entries describes a linear transformation of $\mathbb R^n$. It maps the unit $n$-cube to an $n$-dimensional parallelotope (a parallelogram if $n=2$; a parallelepiped if $n=3$). The magnitude of the determinant is the $n$-dimensional volume of this parallelotope; thus, it describes the $n$-dimensional "volume scaling factor" of the linear transformation. The sign of the determinant indicates whether the transformation preserves or reverses orientation. If the determinant is zero, then the image parallelotope has volume zero and dimension less than $n$, so the linear transformation is not invertible.

J. W. Tanner
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