- #1

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- sorry. erm.. what does the determinant means or functions in matrices , math? thanks..

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- Thread starter yuenkf
- Start date

- #1

- 10

- 0

- sorry. erm.. what does the determinant means or functions in matrices , math? thanks..

- #2

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Define a square matrix(just for simplicity) A with a

det(A)=(a

Which you could also write with two vertical lines, like the abs value.

- #3

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why we want to get the determinant?

- #4

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- #5

Fredrik

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Gold Member

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There's a theorem that says that the following statements about an arbitrary ##n\times n##-matrix ##A## are equivalent:why we want to get the determinant?

(a) ##\det A\neq 0##.

(b) A is invertible

(c) The rows of A are linearly independent elements of ##\mathbb R^n##.

(d) The columns of A are linearly independent elements of ##\mathbb R^n##.

So if you're interested in finding out if any of the statements (b)-(d) is true, you just compute ##\det A##.

The case n=2 is easy to prove. If you define a,b,c,d by ##A=\begin{pmatrix}a & b\\ c & d\end{pmatrix}## you can show that if ##A## has an inverse, it has to be

$$\frac{1}{ad-bc}\begin{pmatrix}d & -b\\ -c & a\end{pmatrix}.$$ So if ##ad-bc## (the determinant of A) is zero, then ##A## isn't invertible.

- #6

Stephen Tashi

Science Advisor

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why we want to get the determinant?

In applications, the determinant can used (in 2-D) to compute the "oriented" area of a parallelogram whose sides are given by 2 row vectors. In higher dimensions, it can be used to compute an "oriented" volume.

Thinking a square matrix M applied as a linear transformation to a vector x, T(x) = Mx, the determinant of M indicates how T expands or contracts volumes (when T is applied to each vector defining a side of a pareallelepiped to produce a transformed parallelepiped).

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