Finding the inverse matrix requires skills in handling matrices, in particular, the ability to calculate the determinant and transpose.
The inverse matrix is found from the elements of the originalBy the formula: A ^ -1 = A * / detA, where A * is the adjoint matrix, detA is the determinant of the original matrix. An attached matrix is a transposed matrix of additions to the elements of the original matrix.
First, find the determinant of the matrix, itMust be different from zero, since further the determinant will be used as a divisor. Suppose, for example, a square matrix of the third order (consisting of three rows and three columns) is given. As you can see, the determinant of our matrix is not equal to zero, therefore there is an inverse matrix.
Find the complements to each element of the matrix A. The complement to A [i, j] is the determinant of the submatrix obtained from the initial deletion of the i-th row and the j-th column, and this determinant is taken with the sign. The sign is determined by multiplying the determinant by (-1) to the power i + j. Thus, for example, the addition to A [2,1] will be the determinant considered in the figure. The sign was as follows: (-1) ^ (2 + 1) = -1.
As a result, you will receive Matrix Add, now transpose it. Transpose is an operation that is symmetric with respect to the main diagonal of the matrix, columns and rows are interchanged. So, you found an affiliate Matrix A *.
Now divide each element by the determinant of the original matrix and get Matrix The inverse of the original.