Determinant and Inverse of Matrices

Determinant and Inverse of Matrices

Finding the inverse of a matrix is very important in many areas of science. For example, decrypting a coded message uses the inverse of a matrix. Determinant may be used to answer this problem. Indeed, let A be a square matrix. We know that A is invertible if and only if . Also if A has order n, then the cofactor A_i,j is defined as the determinant of the square matrix of order (n-1) obtained from A by removing the row number i and the column number j multiplied by (-1)^i+j. Recall

for any fixed i, and

for any fixed j. Define the adjoint of A, denoted adj(A), to be the transpose of the matrix whose ij^th entry is A_ij.

Example. Let

We have

Let us evaluate . We have

Note that . Therefore, we have

Is this formula only true for this matrix, or does a similar formula exist for any square matrix? In fact, we do have a similar formula.

Theorem. For any square matrix A of order n, we have

In particular, if , then

For a square matrix of order 2, we have

which gives

This is a formula which we used on a previous page.