91Ó°ÊÓ

To prove that there exists a sequence of elementary row operations of types 1 and 3 that transforms an \(m \times n\) matrix \(A\) into an upper triangular matrix, we can follow these steps: 1. Start with the given matrix \(A\). 2. Find the first nonzero entry from the top-down in the left-most column (pivot). Swap rows if necessary to place the pivot in the first row (type 1 row operation). 3. Apply type 3 row operations to eliminate entries below the pivot by adding or subtracting multiples of the pivot row. 4. Ignore the first row and first column, and treat the remaining \((m-1) \times (n-1)\) submatrix as a new matrix. Repeat steps 2-3 (search for a pivot, swap rows, eliminate entries below the pivot) for this submatrix. 5. Continue this process recursively, reducing the submatrix size by one row and one column each time, until you reach the smallest size (1 × 1 or 0 × 0). 6. The resulting matrix is an upper triangular matrix formed by applying a sequence of type 1 and type 3 row operations on the original matrix \(A\). Thus, the existence of such a sequence of row operations is proven.