Why do determinants have their particular form? I know that for a matrix A, if det(A)=0 then the matrix does not have an inverse, and hence the associated system of equations does not have a unique solution. However, why do the determinant formulas have the form they do? Why all the complicated co-factor expansions and alternating signs ?To sum it up: I know what determinants do, but its unclear to me why. Is there an intuitive explanation that can be attached to a co-factor expansion??..

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Raphael Singleton · Accepted Answer

Two exercises that may give you the answer you need (no work, no gain):1. Assume you have a square   [  0  ,  1  ]  ×  [  0  ,  1  ] in the   (  x  ,  y  )-plane. Assume for some reason you need to change the variables you are using. The new variables you are using are now   w  =  a  x  +  b  y and   z  =  c  x  +  d  y, where a, b, c and d are numbers. What is the area of the original square under the new coordinate system, the   (  w  ,  z  ) -plane? 2. A multi-linear mapping in             R        2   (bilinear in this case) is a function,   M  :            R        2    ×            R        2    →      R   such that    M  (  a  x  +  b            x      ^        ,  y  )  =  a  M  (  x  ,  y  )  +  b  M  (            x      ^        ,  y  ) and   M  (  x  ,  a  y  +  b            y      ^        )  =  a  M  (  x  ,  y  )  +  b  M  (  x  ,            y      ^        ). The map is alternating if   M  (  x  ,  y  )  =  −  M  (  y  ,  x  ). These two properties are very useful. Exercise: Show that if  has these properties then   M  (  x  ,  y  )  =  k  ⋅  d  e  t      (                                        x            1                                                y            1                                                            x            2                                                y            2                                )

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