Find Reduced Row Echelon Form 2x-3i=[[7,4],[6,5]]

Problem

2*x−3*I=[[7,4],[6,5]]

Solution

1. Identify the identity matrix I for a 2×2 system, which is [[1,0],[0,1]]

2. Substitute the identity matrix into the equation to isolate the term involving x

2*x−3*[[1,0],[0,1]]=[[7,4],[6,5]]

3. Distribute the scalar −3 into the identity matrix.

2*x−[[3,0],[0,3]]=[[7,4],[6,5]]

4. Add the matrix [[3,0],[0,3]] to both sides of the equation.

2*x=[[7,4],[6,5]]+[[3,0],[0,3]]

5. Perform matrix addition by adding corresponding elements.

2*x=[[10,4],[6,8]]

6. Divide the entire matrix by 2 to solve for x

x=[[5,2],[3,4]]

7. Apply Gaussian elimination to find the Reduced Row Echelon Form (RREF) of matrix x Divide the first row by 5

[[1,2/5],[3,4]]

8. Eliminate the first element of the second row by subtracting 3 times the first row from the second row.

[[1,2/5],[0,4−6/5]]

9. Simplify the fraction in the second row.

[[1,2/5],[0,14/5]]

10. Normalize the second row by multiplying it by 5/14

[[1,2/5],[0,1]]

11. Eliminate the second element of the first row by subtracting 2/5 times the second row from the first row.

[[1,0],[0,1]]

Final Answer

RREF(x)=[[1,0],[0,1]]

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