Simplify the Matrix [[a,0,b,2],[a,a,4,4],[0,a,2,b]]

Problem

[[a,0,b,2],[a,a,4,4],[0,a,2,b]]

Solution

1. Identify the goal to simplify the matrix by performing elementary row operations to reach row-echelon form.

2. Eliminate the entry in the second row, first column by replacing (R_2) with (R_2)−(R_1)

[[a,0,b,2],[0,a,4−b,2],[0,a,2,b]]

3. Eliminate the entry in the third row, second column by replacing (R_3) with (R_3)−(R_2)

[[a,0,b,2],[0,a,4−b,2],[0,0,b−2,b−2]]

4. Simplify the third row by dividing (R_3) by b−2 assuming b≠2

[[a,0,b,2],[0,a,4−b,2],[0,0,1,1]]

5. Eliminate the entries in the third column above the pivot by replacing (R_2) with (R_2)−(4−b)*(R_3) and (R_1) with (R_1)−b*(R_3)

[[a,0,0,2−b],[0,a,0,b−2],[0,0,1,1]]

6. Normalize the first and second rows by dividing by a assuming a≠0

[[1,0,0,(2−b)/a],[0,1,0,(b−2)/a],[0,0,1,1]]

Final Answer

[[a,0,b,2],[a,a,4,4],[0,a,2,b]]⇒[[1,0,0,(2−b)/a],[0,1,0,(b−2)/a],[0,0,1,1]]

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