Find the Determinant

Problem

det(16)

Solution

1. Expand along the fourth row to take advantage of the two zeros in that row.

2. Apply the cofactor expansion formula for the determinant of a 4×4 matrix A

det(A)=(a_41)*(C_41)+(a_42)*(C_42)+(a_43)*(C_43)+(a_44)*(C_44)

3. Calculate the first term using the element (a_41)=40

(a_41)*(C_41)=40⋅(−1)(4+1)*det(2)

4. Evaluate the 3×3 determinant for the first term using Sarrus' rule or expansion.

det(2)=2*(80−48)−3*(32−120)+40*(256−1600)

2*(32)−3*(−88)+40*(−1344)=64+264−53760=−53432

5. Multiply by the coefficient for the first part of the expansion.

−40⋅(−53432)=2137280

6. Calculate the second term using the element (a_44)=16

(a_44)*(C_44)=16⋅(−1)(4+4)*det(16)

7. Evaluate the 3×3 determinant for the second term.

det(16)=16*(256−1600)−2*(32−120)+3*(80−48)

16*(−1344)−2*(−88)+3*(32)=−21504+176+96=−21232

8. Multiply by the coefficient for the second part of the expansion.

16⋅(−21232)=−339712

9. Sum the results to find the final determinant.

2137280−339712=1797568

Final Answer

det(16)=1797568

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