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Definition 6. A square matrix is said to be non-degenerate, if it has nonzero determinant

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Consider the identity matrix

.

Definition 7. The inverse matrix for А is the matrix А –1for which

или .

Inverse matrices exist only for non-degenerate matrices.

The method for finding the inverse matrix. Consider a square non-degenerate matrix of size n n

.

Let us compose the square matrix

,

where the Аij are the algebraic complements of аij. The matrix А * is called the adjoint matrix.

Let us multiply А by А * :

The remaining elements of the first row in the new matrix are the sum of the products of the elements of the first row and the algebraic complements of the parallel rows. According to property 10, they equal zero. The same is true for the elements of the second row, except the second element, which are the sum of the products of the elements of the second row by the algebraic complements of the first, third,... rows.

To obtain the identity matrix from this diagonal matrix, we must divide it by the determinant , i.e., ; consequently, , ,

.

 


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