» Required Reading » Table of Contents » Chapter 4 4.4 The Determinant and the Inverse of a Matrix
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The inverse of a .square matrix M is a matrix M^-1 with the property that
M^-1 M = M M^-1  = I, where I is the identity matrix of the same size, having 1’s on the diagonal and 0’s elsewhere. In terms of transformations, M^-1 undoes the transformation produced by M.
The condition MM^-1 = I can be written as

.

These are the same as the two conditions (A) and (B) of section 4.3 except that det M is on the lefthand side in (A), and (-1)^i+jM_ij appears in them instead of M^-1_ji.
We can therefore deduce from the last previous results:

When the determinant of a matrix is 0 the inverse matrix will not exist, and as we have noted, there must be a linear combination of the columns that is zero and a linear combination of the rows that is zero. The result of multiplying a matrix by a column vector is to give the linear combination of its columns defined by the components of the vector. We can deduce that there is a non-trivial column vector which the matrix takes into the 0 column vector, and similarly for row vectors.