Step-by-step explanation:
To transform a matrix into its normal form, you can perform row operations on it until it is in row-reduced echelon form. In row-reduced echelon form, the matrix will have the following properties:
1. The first non-zero element in each row is called the pivot, and it is always a 1.
2. The pivot is always the leftmost non-zero element in its row.
3. Each pivot is strict to the right of the pivot in the row above it.
4. All elements below the pivot are zero.
To transform a matrix into its normal form, you can perform the following row operations:
1. Swap two rows.
2. Multiply a row by a non-zero constant.
3. Add a multiple of one row to another row.
Using these row operations, we can transform the given matrix into its normal form:
1. Swap rows 1 and 2:
2 -5 1 2
1 -2 3 -2
3 8 6 2
5 -12 -1 6
2. Subtract 3 times row 2 from row 1:
-1 -11 -2 6
1 -2 3 -2
3 8 6 2
5 -12 -1 6
3. Subtract 5 times row 3 from row 4:
-1 -11 -2 6
1 -2 3 -2
3 8 6 2
0 4 -7 0
4. Divide row 3 by 3:
-1 -11 -2 6
1 -2 3 -2
1 2 2 2
0 4 -7 0
5. Subtract row 3 from row 1:
0 -13 -4 4
1 -2 3 -2
1 2 2 2
0 4 -7 0
6. Divide row 2 by -2:
0 -13 -4 4
-1 1 -1.5 1
1 2 2 2
0 4 -7 0
7. Subtract row 2 from row 3:
0 -13 -4 4
-1 1 -1.5 1
0 1 0.5 1
0 4 -7 0
8. Subtract row 3 from row 4:
0 -13 -4 4
-1 1 -1.5 1
0 1 0.5 1
0 3 -8 -1
9. Subtract 3 times row 4 from row 1:
0 -16 -13 1
-1 1 -1.5 1
0 1 0.5 1
0 3 -8 -1
10. Divide row 1 by -16:
0 1 0.81 -0.0625
-1 1 -1.5 1
0 1 0.5 1
0 3 -8 -1
After these row operations, the matrix is in row-reduced echelon form, which is its normal form. The normal form of the matrix is:
0 1 0.81 -0.0625
0 0 0 0
0 0 0 0
0 0 0 0