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
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
For a square matrix of order 2, we have
On the next page, we will discuss the application of the above formulas to linear systems.
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Author: M.A. Khamsi