Find the Determinant

Problem

det(−2)

Solution

1. Perform row operations to create zeros in the first column. Use the third row as a pivot by swapping it with the first row. This changes the sign of the determinant.

det(A)=−det(1)

2. Eliminate the entries below the pivot in the first column using the operations (R_2)→(R_2)−3*(R_1) (R_3)→(R_3)+2*(R_1) and (R_4)→(R_4)−2*(R_1)

det(A)=−det(1)

3. Expand along the first column to reduce the matrix to a 3×3 determinant.

det(A)=−1⋅det(8)

4. Simplify the 3×3 matrix by adding the second row to the first row ((R_1)→(R_1)+(R_2).

det(A)=−det(1)

5. Create another zero in the first row by adding the first column to the second column ((C_2)→(C_2)+(C_1).

det(A)=−det(1)

6. Expand along the first row and calculate the resulting 2×2 determinant.

det(A)=−1⋅((6⋅9)−(−11⋅−6))

7. Calculate the final arithmetic value.

det(A)=−1⋅(54−66)

det(A)=−1⋅(−12)

det(A)=12

Final Answer

det(−2)=12

Want more problems? Check here!