Find the Determinant

Problem

det(2)

Solution

1. Perform row operations to simplify the matrix. Subtract the first row ((R_1) from the second row ((R_2) and the fourth row ((R_4).

(R_2)→(R_2)−(R_1)

(R_4)→(R_4)−(R_1)

det(2)

2. Perform another row operation to create a zero in the first column of the third row ((R_3). Subtract twice the first row from the third row.

(R_3)→(R_3)−2*(R_1)

det(2)

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

2⋅det(0)

4. Expand the 3×3 determinant along the first row.

2⋅(0⋅det(−6)−3⋅det(76)+1⋅det(76))

5. Calculate the 2×2 determinants inside the expansion.

det(76)=(76)*(−3)−(−9)*(20)=−228+180=−48

det(76)=(76)*(−1)−(−6)*(20)=−76+120=44

6. Substitute the values back into the expansion expression.

2⋅(−3*(−48)+1*(44))

2⋅(144+44)

2⋅188=376

Final Answer

det(2)=376

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