Answer:
To find the matrix U such that B = UA we can start by performing row operations on matrix A to reduce it to its row-echelon form (RREF which will give us matrix B.
Given matrix A:
A = 1 3 -5
0 -2 5
3 -1 6
We can perform row operations on A to reduce it to RREF:
1. Multiply row 2 by 3 and add it to row 1:
A' = 1 3 -5
0 -2 5
0 8 -4
2. Multiply row 3 by -1/2 and add it to row 2:
A' = 1 3 -5
0 -2 5
0 0 3
3. Multiply row 3 by 1/3:
A' = 1 3 -5
0 -2 5
0 0 1
This is the row-echelon form B. Now we can find the matrix U such that B = UA.
Let U be the matrix of row operations performed on A to obtain B:
U = 1 0 0
0 1 0
-3 1 1/3
To verify that B = UA let's multiply U by A:
UA = 1 0 0 * 1 3 -5 = 1 3 -5
0 1 0 0 -2 5 0 -2 5
-3 1 1/3 3 -1 6 0 0 1
As we can see B = UA.