To find the reduced row-echelon form
�
�
A
R
of matrix
�
A, we can use row operations. Here are the steps:
Step 1: Perform row operations to create zeros below the leading coefficient of the first row:
R2 = R2 - (1/3)R1
R3 = R3 - (4/3)R1
The matrix after Step 1 becomes:
(
3
2
−
12
0
−
5
3
5
0
−
5
3
5
)
⎝
⎛
3
0
0
2
−
3
5
−
3
5
−12
5
5
⎠
⎞
Step 2: Perform row operations to create a leading coefficient of 1 in the second row:
R2 = -\frac{3}{5}R2
The matrix after Step 2 becomes:
(
3
2
−
12
0
1
−
1
0
−
5
3
5
)
⎝
⎛
3
0
0
2
1
−
3
5
−12
−1
5
⎠
⎞
Step 3: Perform row operations to create zeros above and below the leading coefficient of the second row:
R1 = R1 - 2R2
R3 = R3 + \frac{5}{3}R2
The matrix after Step 3 becomes:
(
3
0
−
10
0
1
−
1
0
0
0
)
⎝
⎛
3
0
0
0
1
0
−10
−1
0
⎠
⎞
Step 4: Perform row operations to create a leading coefficient of 1 in the first row:
R1 = \frac{1}{3}R1
The matrix after Step 4 becomes:
(
1
0
−
10
3
0
1
−
1
0
0
0
)
⎝
⎛
1
0
0
0
1
0
−
3
10
−1
0
⎠
⎞
The reduced row-echelon form
�
�
A
R
of matrix
�
A is:
�
�
=
(
1
0
−
10
3
0
1
−
1
0
0
0
)
A
R
=
⎝
⎛
1
0
0
0
1
0
−
3
10
−1
0
⎠
⎞
To find the matrix
Ω
Ω such that
Ω
�
=
�
�
ΩA=A
R
, we perform the same row operations on the identity matrix:
Ω
=
(
1
3
0
0
0
−
3
5
0
0
5
3
1
)
Ω=
⎝
⎛
3
1
0
0
0
−
5
3
3
5
0
0
1
⎠
⎞
Therefore,
Ω
�
=
�
�
ΩA=A
R