Both row full rank and column full rank are n-order matrices, that is, n-order matrices. Row full rank matrix has nothing to do with row vector linearity, and column full rank matrix has nothing to do with column vector linearity. So if it is a square matrix, the row full rank matrix and the column full rank matrix are equivalent.

In the multiplication of matrices, there is a kind of matrix that plays a special role, just like 1 in the multiplication of numbers. We call this matrix identity matrix, or identity matrix for short. It is a square matrix, and the elements on the diagonal from the upper left corner to the lower right corner (called the main diagonal) are all 0 except 1.

Linear equations can be solved by transforming the coefficient matrix into identity matrix.

theorem

Uniqueness of (1) inverse matrix. If the matrix A is invertible, the inverse matrix of A is unique, and the inverse matrix of A is A- 1.

(2) The necessary and sufficient condition for the invertibility of N-order square matrix A is r (a) = m ... For N-order square matrix A, if r(A)=n, then A is called a full rank matrix or a nonsingular matrix.

(3) Any full rank matrix can be transformed into identity matrix by finite elementary row transformation. It is deduced that the inverse matrix A of full rank matrix A can be expressed as the product of finite elementary matrices.