Matrix Inverse | Research & Encyclopedia Articles

Matrix Inverse

If an n x n matrix A has a nonzero determinant, then there is a matrix called "A inverse", denoted by A^-1 such that AA^-1 = A^-1A = Id. Here Id denotes the identity matrix, that is the matrix that has ones on its diagonal entries and zeros everywhere else. If there is matrix B, say, that has the property that BA = AB = Id then B = A^-1. Also, the determinant of A is nonzero because the determinant of a product of matrices is equal to the product of the matrices' determinants. Hence det(A)det(B) = det (Id) = 1. So, det(A) = 1/det(A^-1).

The entries of A^-1 can be found with Cramer's rule. Since AA^-1 = Id, if the jth column of A^-1 is denoted by A^-1_j then the jth column of Id is equal to AA^-1_j. By Cramer's rule, the kth entry of the jth column of A is equal the determinant of A_k(e_j) divided by the determinant of A. Here e_j denotes the jth column of Id. The determinant of A_k(e_j) is equal to (-1)^j+k times the determinant of the matrix A(j,k) which is obtained from A by removing the jth row and the kth column. The number equal to (-1)^j+k times the det A(j,k) is called the (j,k)-cofactor of A.

Here is a method for finding the determinant of the matrix A. First pick a column or a row. If the chosen column or row has a lot of zeros, the determinant will be easier to calculate. Suppose that the jth row is the chosen one. Then the determinant is equal to the sum from k = 1 to n of the (j,k)-entry of A times C(j,k). This definition of determinant does not depend on the row or column chosen.

This fact can be proven using elementary matrices (for definitions see the article Matrix multiplication). A is equal to a product of the form BE₁...E_m in E_i is an elementary matrix for each i and B has all of its nondiagonal entries equal to zero and all of its diagonal entries equal to either one or zero. The proof that A equals this uses Gaussian elimination (see the article on Systems of Linear Equations). Gaussian elimination implies that any matrix, A in particular, can be multiplied by a sequence elementary matrices to result in an upper triangular matrix. That is a matrix with all of its (i,j) entries equal to zero whenever i > j. But, Gaussian elimination can be continued by exchanging the words "row" and "column". After this continuation, the result is a matrix B that has the required form. It is easy to prove that for the matrix B, its determinant does not depend on the row or column chosen for cofactor expansion. By induction, we assume that there is a number N 0 such that if m is N then any matrix M of the form M = BE₁...E_m has the property that its determinant does not depend on the particular row or column chosen for cofactor expansion. It is easy to show that for any elementary matrix E, the product ME also has this property. By induction, therefore, it is true that the determinant of any matrix does not depend on the particular row or column chosen.