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I am teaching 9th graders about matrices and looking for an explanation of what a determinant is. All of the explanations I find are too complex! Is there someone that can give a 9th grade level explanation??

Teri
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    I think the best way would be to show example of 2x2, 3x3 systems where the determinant shows up in the solution. Could add that the determinant is a function that takes as input a matrix and yields a number. But have they covered the concept of functions? – Peter Grill Apr 28 '13 at 00:30
  • If $p$ is a polynomial and $x$ is the matrix and $p(x)=0$ , then the absolute value of the determinant is the same as the absolute value of one of the complex numbers $z$ such that $p(z)=0$. – mick Apr 28 '13 at 00:37
  • @mick, are you saying that one of the eigenvalues of a matrix has absolute value equal to that of the matrix's determinant? Perhaps you mean product of the complex roots, but then you need to allow multiplicity of roots which means you need to be using the characteristic polynomial (which is defined in terms of determinants), or at the very least using the minimal polynomial and then scale as necessary if its degree is not the same as the dimension of the matrix. Either way, I do not find that a very enlightening way of thinking about determinants, although it is a useful perspective at times. – anon Apr 28 '13 at 00:48

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Determinant is $\pm$ the area of the parallelogram in 2 dimensions, $\pm$ the volume of the parallelepiped in 3 dimensions.

vadim123
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    And you can demonstrate a proof by elemantary basis transformations, starting out from the unit square $\pmatrix{1&0\0&1}$, arriving to the given $2\times 2$ matrix, using steps $e_i:=e_i+\lambda e_j$, $\ e_i:=\gamma e_i$ (for $\gamma\ne 0$) and $e_i\leftrightarrow e_j$. – Berci Apr 28 '13 at 02:09