Find the Determinant

Problem

det(3)

Solution

1. Factor out the constant 4 from the second row to simplify the matrix.

det(A)=4⋅det(3)

2. Perform row operations to create zeros in the second row. Subtract the first column from the second, third, and fourth columns ((C_2)−(C_1) (C_3)−(C_1) (C_4)−(C_1).

det(A)=4⋅det(3)

3. Expand along the second row using the cofactor expansion method. The element (a_2,1)=1 is in a position where the sign is negative ((−1)(2+1).

det(A)=4⋅(−1)⋅(1)⋅det(1)

4. Simplify the 3x3 determinant by performing a column operation. Add −2 times the second column to the third column ((C_3)−2*(C_2) to create a zero in the third row.

det(A)=−4⋅det(1)

5. Expand along the third row of the 3x3 matrix. The element (a_3,2)=−1 is in a position where the sign is negative ((−1)(3+2).

det(A)=−4⋅(−1)⋅(−1)⋅det(1)

6. Calculate the 2x2 determinant using the formula a*d−b*c

det(A)=−4⋅(1⋅400−39⋅343202)

7. Multiply and subtract the values inside the parentheses.

39⋅343202=13384878

400−13384878=−13384478

8. Multiply by the leading coefficient to find the final result.

−4⋅(−13384478)=53537912

Final Answer

det(3)=53537912

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