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I guess I am having a little trouble understanding how or why the determinant was developed to begin with. I am using it in a Calculus 3 lecture although used it is not discussed in the context of linear algebra. Is there some "obvious" geometry I miss and although beyond the scope of the original question , can I assume the 3 by 3 case can be gotten by induction in some way? Thank you.

Sedumjoy
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4 Answers4

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The determinant of a linear map $T: \mathbb{R}^n \to \mathbb{R}^m$ is the oriented volume of the parallelepiped spanned by the column vectors of the associated matrix to $T$.

Faraad Armwood
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    This is a particularly important property in the context of algebraic number theory, where the determinant is used to measure the sizes of lattices – Stella Biderman Mar 23 '17 at 17:56
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The property of having a non-zero determinant is extremely important. In some sense (see another answer about parallelepiped), the determinant measures the "size" of a matrix. Matrices with size $0$ have some very important properties, as seen in the invertable matrix theorem. Probably the most important of these is that a matrix has a multiplicative inverse if and only if it has non-zero determinant.

Stella Biderman
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"Obvious" is subjective and personally I wouldn't call any practical application of matrices obvious.

But one application is computer graphics/text.

You can use matrices to do what's called a shear transformation. This is helpful for turning standard text into italicized text. Boring, I know, but this technique can also be applied to images.

EDIT: Your actual question appears to be about determinants and not about applications of $2 \times 2$ matrices. (?)

Historically, determinants were used as properties of linear systems of equations to help analyze solutions. Specifically, they were useful in figuring out whether or not a system had a unique solution. See here for more info.

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If you take, for example starting vectors [3 0] and [0 3]; the square [3 0] x [0 3] has area of 9. Now let's suppose your matrix M is \begin{bmatrix}-2 & 0 \\0 & -2\end{bmatrix} with a determinant of 4. If we multiply the starting vectors with your matrix we get:

\begin{align}\begin{bmatrix}-2 & 0 \\0 & -2\end{bmatrix}\begin{bmatrix}3 & 0 \\0 & 3\end{bmatrix} = \begin{bmatrix}-6 & 0 \\0 & -6\end{bmatrix}\end{align} the square [-6 0] x [0 -6] has area 36.

The original area was stretched by 4 (the determinant of your matrix).

The determinant tells you the factor of area or volume change of the transformation.

Blix
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