Find the Determinant

Problem

det(43)

Solution

1. Perform row operations to simplify the matrix. Subtract the first row from the third row ((R_3)→(R_3)−(R_1) to create zeros or smaller numbers.

|[43,46,55,47],[42,55,45,69],[0,−4,6,5],[47,41,63,0]|

2. Perform row operations to further reduce the first column. Subtract the first row from the second row ((R_2)→(R_2)−(R_1).

|[43,46,55,47],[−1,9,−10,22],[0,−4,6,5],[47,41,63,0]|

3. Perform row operations to eliminate the value in the fourth row, first column. Add 47 times the second row to the fourth row ((R_4)→(R_4)+47*(R_2).

|[43,46,55,47],[−1,9,−10,22],[0,−4,6,5],[0,464,−407,1034]|

4. Perform row operations to eliminate the value in the first row, first column. Add 43 times the second row to the first row ((R_1)→(R_1)+43*(R_2).

|[0,433,−375,993],[−1,9,−10,22],[0,−4,6,5],[0,464,−407,1034]|

5. Expand along the first column using the cofactor method. Since there is only one non-zero entry in the first column at position (2,1) the determinant is (−1)⋅(−1)(2+1) times the 3×3 minor.

1⋅|[433,−375,993],[−4,6,5],[464,−407,1034]|

6. Simplify the 3×3 matrix using row operations. Add 116 times the second row to the third row ((R_3)→(R_3)+116*(R_2).

|[433,−375,993],[−4,6,5],[0,289,1614]|

7. Expand the 3×3 determinant using the rule of Sarrus or cofactor expansion along the first column.

433*(6⋅1614−5⋅289)−(−4)*(−375⋅1614−993⋅289)

8. Calculate the numerical values of the products.

433*(9684−1445)+4*(−605250−286977)

9. Combine the results to find the final value.

433*(8239)+4*(−892227)

3567487−3568908=−1421

Final Answer

det(43)=−1421

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