The inverse matrix of the third-order matrix is calculated as follows:
First, we need to make sure that the given third-order matrix is reversible. The condition for a matrix to be reversible is that its determinant is not 0. If the determinant is 0, then this matrix is irreversible and there is no inverse matrix.
If the matrix is invertible, we can use Gaussian elimination to solve its inverse matrix. The specific steps are as follows:
a.Represent a given third-order matrix as an augmented matrix, i.e., add an identity matrix of the same shape as the original matrix to the right of the original matrix.
b.The Gaussian-approximate elimination method is used to reduce the augmented matrix to the line minimalist form. In this process, we change each column of the original matrix to a multiple of the row it is in, while keeping the value of the determinant unchanged.
c.When the augmentation matrix is reduced to the simplest form of the row, we find that the last element of the last row is a non-zero number, which is the determinant value of the original matrix. And the penultimate element of the last row is 1.
d.Finally, we divide the penultimate element and the penultimate element of the last row by the determinant values respectively to get the inverse matrix of the original matrix.
It should be noted that this method only works with third-order phalanxes. For higher-order matrices, the method of solving the inverse matrix is more complex.