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

for any fixed

**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

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