Find the Determinant

Problem

det(−9/7)

Solution

1. Factor out the common denominator 1/21 from each row to simplify the matrix. Since there are 3 rows, we pull out (1/21)3

det(A)=(1/21)3*det(−27)

2. Simplify the matrix further by factoring out constants from specific rows or columns. Notice that in the first row, we can factor out 3, and in the third row, we can factor out 3.

det(A)=(3⋅3)/21*det(−9)

3. Apply row operations to create zeros. Add 1 times the second row to the third row ((R_3)→(R_3)+(R_2).

det(A)=9/9261*det(−9)

4. Apply another row operation. Add 6 times the first row to the second row ((R_2)→(R_2)+6*(R_1).

det(A)=1/1029*det(−9)

5. Expand along the second column, which contains two zeros.

det(A)=1/1029⋅(−1)⋅(1)*det(−14)

6. Calculate the 2×2 determinant.

det(−14)=(−14)*(−7)−(−7)*(−7)

98−49=49

7. Multiply by the external scalar.

det(A)=−1/1029⋅49

−49/1029=−1/21

Final Answer

det(−9/7)=−1/21

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