Simplify the Matrix

Problem

[[5,6,0,1],[6,6,0,0],[6,5,6,9],[56,35,1,0]]

Solution

1. Identify the task as finding the determinant of the given 4×4 matrix, as "simplify" for a square matrix often refers to evaluating its determinant or reducing it to row-echelon form. We will calculate the determinant.

2. Expand along the second row because it contains two zeros, which simplifies the calculation.

|A|=−6*|[6,0,1],[5,6,9],[35,1,0]|+6*|[5,0,1],[6,6,9],[56,1,0]|

3. Evaluate the first 3×3 determinant using expansion along the first row.

|[6,0,1],[5,6,9],[35,1,0]|=6*(0−9)−0+1*(5−210)

|[6,0,1],[5,6,9],[35,1,0]|=−54−205=−259

4. Evaluate the second 3×3 determinant using expansion along the first row.

|[5,0,1],[6,6,9],[56,1,0]|=5*(0−9)−0+1*(6−336)

|[5,0,1],[6,6,9],[56,1,0]|=−45−330=−375

5. Combine the results into the original expansion formula.

|A|=−6*(−259)+6*(−375)

|A|=1554−2250

|A|=−696

Final Answer

det(5)=−696

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