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Matrices math

  1. Dec 25, 2014 #1
    1. sorry. erm.. what does the determinant means or functions in matrices , math? thanks..
     
  2. jcsd
  3. Dec 25, 2014 #2
    Example:
    Define a square matrix(just for simplicity) A with a1 and b1 in the top row and and a2 and b2 in the second row and a in the first columm and bs in the second columm.The determinant of the matrix would be
    det(A)=(a1*b2-b1*a2)
    Which you could also write with two vertical lines, like the abs value.
     
  4. Dec 25, 2014 #3
    why we want to get the determinant?
     
  5. Dec 25, 2014 #4
    One can show that linear equation sytems have solutions for exapmle. It comes n handy all the time. Another example would be that one can write many equation in quantum mechanocs much more elegant that way.
     
  6. Dec 25, 2014 #5

    Fredrik

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    There's a theorem that says that the following statements about an arbitrary ##n\times n##-matrix ##A## are equivalent:

    (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.
     
  7. Dec 25, 2014 #6

    Stephen Tashi

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    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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