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Two matrices are said to be row equivalent if one can be obtained from the other using a finite sequence of row operations. There are three types of row operations that can be performed to a matrix: 1. Swapping two rows 2. Multiplying a row by a nonzero scalar 3. Adding a multiple of one row to another row

Let's assume matrix A is row equivalent to matrix B, which means we can get matrix B from matrix A using a finite sequence of row operations. We also assume that matrix B is row equivalent to matrix C, meaning that we can get matrix C from matrix B using a finite sequence of row operations. Now we want to prove that matrix A is row equivalent to matrix C. To do this, we need to show that we can get matrix C from matrix A using a finite sequence of row operations. Since we can get matrix B from matrix A using a finite sequence of row operations and matrix C from matrix B using another finite sequence of row operations, we can combine these two sequences of operations to get matrix C from matrix A. Thus, row equivalence is transitive, and matrix A is row equivalent to matrix C. #Part b:

A matrix is said to be in reduced row echelon form (RREF) when it meets specific criteria, including that the pivots (or leading entries) are 1 and located to the right of any pivots in previous rows, and all nonzero rows are above any rows of all zeros. A nonsingular matrix is a matrix with a nonzero determinant. It is also invertible, meaning that it has an inverse matrix. Nonsingular matrices have unique RREF because they have full rank, which means that there are no linearly dependent rows.

Let matrix A and matrix B be two nonsingular n x n matrices. We want to prove that these matrices are row equivalent. Since both matrix A and matrix B are nonsingular, they have unique RREF. Moreover, for any n x n nonsingular matrix, its RREF is the identity matrix of size n x n. Therefore, both matrix A and matrix B can be transformed into the n x n identity matrix using a finite sequence of row operations. Since matrix A can be transformed into the identity matrix using a finite sequence of row operations and matrix B can also be transformed into the identity matrix using another finite sequence of row operations, we can combine these two sequences of operations to get from matrix A to matrix B (by first transforming A to the identity matrix and then transforming the identity matrix to B). Thus, any two nonsingular n x n matrices are row equivalent.