Algebra
1.We say that two matrices A and B are row-equivalent if one can be obtained from the other by a finite sequence of elementary row operation. Show that if A and B are row-equivalent matrices, then the homogeneous systems of linear equations Ax = 0 and Bx = 0 have exactly same solutions.
Suppose R and R’ are 2 × 3 row-reduced echelon matrices and that the systems RX = 0 and R’X = 0 have exactly the same solutions. Prove that R = R’
Suppose A is a 2 × 1 matrix and that B is a 1 × 2 matrix. Prove that C = AB is not invertible.
For each matrix use elementary row operations to discover whether it is invertible, and to find the inverse in case it is.
a) 2 5 −1
4 −1 2
6 4 1
b) 1 −1 2
3 2 4
0 1 −2
c) 1 2 3 4
0 2 3 4
0 0 3 4
0 0 0 4
Suppose R and R’ are 2 × 3 row-reduced echelon matrices and that the systems RX = 0 and R’X = 0 have exactly the same solutions. Prove that R = R’.
