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 Ai,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 ijth entry is Aij.

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.

On the next page, we will discuss the application of the above formulas to linear systems.

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Author: M.A. Khamsi