Why does expansion by minors work

It is often most efficient to use a combination of several techniques when computing the determinant of a matrix. Indeed, when expanding cofactors on a matrix, one can compute the determinants of the cofactors in whatever way is most convenient. Or, one can perform row and column operations to clear some entries of a matrix before expanding cofactors, as in the previous example. This is usually most efficient when there is a row or column with several zero entries, or if the matrix has unknown entries.

This is generally the fastest when presented with a large matrix which does not have a row or column with a lot of zeros in it. Cofactor expansion is recursive, but one can compute the determinants of the minors using whatever method is most convenient. Or, you can perform row and column operations to clear some entries of a matrix before expanding cofactors. Recall from this proposition in Section 3. The matrix of cofactors is sometimes called the adjugate matrix of A , and is denoted adj A :.

Note that the i , j cofactor C ij goes in the j , i entry the adjugate matrix, not the i , j entry: the adjugate matrix is the transpose of the cofactor matrix. It is clear from the previous example that 4. However, it has its uses. Let A i be the matrix obtained from A by replacing the i th column by b. Then the matrix A i looks like this:.

This vector is the solution of the matrix equation. Expanding cofactors along the i th column, we see the determinant of A i is exactly the j , i -cofactor C ji of A. Using this terminology, the equation given above for the determinant of the 3 x 3 matrix A is equal to the sum of the products of the entries in the first row and their cofactors:.

This is called the Laplace expansion by the first row. It can also be shown that the determinant is equal to the Laplace expansion by the second row,. Even more is true. The determinant is also equal to the Laplace expansion by the first column.

Although the Laplace expansion formula for the determinant has been explicitly verified only for a 3 x 3 matrix and only for the first row, it can be proved that the determinant of any n x n matrix is equal to the Laplace expansion by any row or any column.

Example 1 : Evaluate the determinant of the following matrix using the Laplace expansion by the second column:. The minors of these entries, mnr a 12 , mnr a 22 , and mnr a 32 , are computed as follows:.

Note that it was unnecessary to compute the minor or the cofactor of the 3, 2 entry in A , since that entry was 0. In general, then, when computing a determinant by the Laplace expansion method, choose the row or column with the most zeros.

Although efficient for small matrices, techniques such as Gaussian elimination are much more efficient when the matrix size becomes large. Let denote the determinant of an matrix , then for any value , For example, for a matrix, the above formula gives. The procedure can then be iteratively applied to calculate the minors in terms of subminors, etc. The factor is sometimes absorbed into the minor as. The equation for the determinant can also be formally written as.

Arfken, G. Mathematical Methods for Physicists, 3rd ed. M 1,1. M 1,2. M 1,3. M 1,4. Stapel, Elizabeth. Accessed [Date] [Month]