Matrix Inversion 2.10 2001/02/16 2002/10/24 Bailey Edwards ulysses@alumni.rice.edu Thanos Antoulas aca@rice.edu John Paul Slavinsky jps@alumni.rice.edu Roy Ha rha@rice.edu Elizabeth Chan lizychan@rice.edu Bailey Edwards ulysses@alumni.rice.edu Thanos Antoulas aca@rice.edu John Paul Slavinsky jps@alumni.rice.edu matrix matrix inversion inverse adjoint determinant (Blank Abstract) Let's say we have the square n x n matrix A composed of real numbers. By "square", we mean it has the same number of rows as columns. A a11 … a1n ⋮ ⋱ ⋮ an1 … ann The subscripts of the real numbers in this matrix denote the row and column numbers, respectively (i.e. a12 holds the position at the intersection of the first row and the second column). We will denote the inverse of this matrix as A . A matrix inverse has the property that when it is multiplied by the original matrix (on the left or on the right), the result will be the identity matrix. A A A A I To compute the inverse of A, two steps are required. Both involve taking determinants ( A ) of matrices. The first step is to find the adjoint ( A ) of the matrix A. It is computed as follows: A α11 … α1n ⋮ ⋱ ⋮ αn1 … αnn αij -1 i j A ij where A ij is the n1 x n1 matrix obtained from A by eliminating its i-th column and j-th row. Note that we are not eliminating the i-th row and j-th column as you might expect. To finish the process of determining the inverse, simply divide the adjoint by the determinant of the original matrix A . A 1 A A A a b c d Find the inverse of the above matrix. The first step is to compute the terms in the adjoint matrix: α11 -1 2 d α12 -1 3 b α21 -1 3 c α22 -1 4 a Therefore, A d b c a We then compute the determinant to be a d b c . Dividing through by this quantity yields the inverse of A: A 1 a d b c d b c a