Determinant of an Invertible Matrix

Date: 08/04/99 at 03:37:14
From: Mike Smith
Subject: Determinants

I've been having problems understanding what determinants have to do 
with matrix invertibility. Could you give me an explanation as well as 
a proof of why a matrix is invertible if and only if the determinant 
is non-zero? I would really like to know.

Mike Smith.

Date: 08/04/99 at 07:54:11
From: Doctor Floor
Subject: Re: Determinants

Dear Mike,

Thanks for your question:

Let us consider the following general nxn matrix:

         [a11 a12 ... a1n]
     M = [a21 a22 ... a2n]
         [ :   :       : ]
         [an1 an2 ... ann]

We define the cofactor of a11, call it c11, by the determinant of the 
matrix found when we leave row 1 and column 1 out of M, so:

           |a22 ... a2n|
     c11 = | :       : |
           |an2 ... ann|

In almost the same way, the cofactor cpq of apq is found as the 
determinant of the matrix when we leave row p and column q out of M. 

Now we have to use the rule of signs:
   If p+q is even then cpq = determinant,
   if p+q is odd then cpq = -determinant.

Doing this we find that:

     ap1*cp1 + ap2*cp2 + ap3*cp3 + ... + apn*cpn = det(M)   .....[1]

for any 1 <= p <= n.

Also, we find that:

     aq1*cp1 + aq2*cp2 + aq3*cp3 + ... + aqn*cpn = 0   ..........[2]

when 1 <= p <= n and 1 <= q <= n, while p and q are unequal.

Expression [2] is the determinant of the matrix you get when in matrix 
M you replace row p by row q. The matrix then has two equal rows, so 
clearly its determinant must be 0.

We find the inverse of M as:

              [c11/det(M) ... cn1/det(M)]
     M^(-1) = [c12/det(M) ... cn2/det(M)]
              [   :              :      ]
              [c1n/det(M) ... cnn/det(M)]

because now M*M^(-1) = I (the nxn identity matrix) (check this).

This computation of the inverse of M clearly can't work when 
det(M) = 0.

The fact that a matrix with determinant 0 really can't have an inverse 
can be seen from the following:

We know that, when det(M) = 0, then row n is a linear combination of 
rows 1, 2, 3, ..., n-1.

Now suppose that matrix N is the inverse of matrix M, so M*N = I (the 
identity matrix). Note that the nth row of M*N is now a linear 
combination of the rows 1, 2, 3, ..., n-1 of M*N too. The elements in 
the nth column of rows 1, 2, 3, ..., n-1 of I are all zero, so that 
means that the nth element of the nth row of M*N should be zero, too, 
being a linear combination of zeroes. But the nth element of the nth 
row of I is one, and we have a contradiction.

I hope this cleared things up. If you have more questions, don't 
hesitate to send them to Dr. Math.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/