Math Is Fun Forum

Janiffer

Member

Registered: 2008-04-26

Posts: 12

Determinants

Suppose A  is a square matrix such that det A^4=0. Esplain why A cannot be invertible.

b) Consider a square invertible matrix A. Ifthe entries of A are integers and det A= +/- 1, then show that the entries of A inverse are integers....:D

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Determinants

Hint for part (a):

If A is invertible, det(A) ≠ 0.

Last edited by JaneFairfax (2008-05-25 21:57:10)

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Determinants

To find the inverse of a matrix A, you divide its adjugate by its determinant.

Its adjugate is found by making its matrix of minors, then it matrix of cofactors, then taking the transpose of that.

A minor corresponds to an element of an nxn matrix. It is equal to the determinant of the (n-1)x(n-1) matrix created by removing the row and column from the original matrix that the element was in.

eg.

X = ( 1 2 3 )
      ( 4 5 6 )
      ( 7 8 9 )

The minor of the first element of this matrix would be det ( 5 6 ) = 45 - 48 = -3.
                                                                                   ( 8 9 )

Finding all the minors of a matrix lets you create the matrix of minors, by putting each minor in its appropriate place.

The matrix of minors for X would be:
( -3  -6  -3 )
( -6 -12 -6 )
( -3  -6  -3 )

And I had no idea that matrix would be so cool when I made up this example. Nifty.

Anyway, next you make the matrix of cofactors. This is made by changing the sign of some of the elements of your matrix of minors, in a checkered pattern. For example, if the top-left square of your matrix was a black square on a chessboard, you'd change the signs of every white square. Another way of thinking about it is that you multiply the element in the ith row and jth column of the matrix by (-1)^(i+j).

Hence, the matrix of cofactors for X would be:
( -3   6 -3 )
(  6 -12 6 )
( -3  6  -3 )

Finally, you take the transpose of this (ie. flip it along its main diagonal). In this case, the matrix is symmetric and so that doesn't change anything.

Finally, you divide every element in that matrix by the determinant of the original matrix and you have your inverse.

To show that the elements in A inverse in your question are integers, just think about every process in this method, and show that if integers are put in, then integers are guaranteed to come out.

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Why did the vector cross the road?
It wanted to be normal.