Find the Determinant

Problem

det(1)

Solution

1. Perform row operations to create zeros in the first column. Replace (R_2) with (R_2)−2*(R_1) (R_3) with (R_3)−4*(R_1) (R_4) with (R_4)−3*(R_1) and (R_5) with (R_5)+5*(R_1)

det(1)

2. Expand along the first column to reduce the matrix to a 4×4 determinant.

1⋅det(4)

3. Simplify the rows to make calculations easier. Replace (R_2) with (R_2)−4*(R_1) and (R_3) with (R_3)−2*(R_1)

det(4)

4. Create zeros in the first column of the 4×4 matrix. Replace (R_1) with (R_1)+4*(R_2) (R_3) with (R_3)−(R_2) and (R_4) with (R_4)−11*(R_2)

det(0)

5. Expand along the first column again. Note the sign for the element at (2,1) is negative.

−(−1)⋅det(69)

6. Factor out constants to simplify the 3×3 matrix. Factor 2 from the second row.

2⋅det(69)

7. Apply row operations to the 3×3 matrix. Replace (R_1) with (R_1)+8*(R_2) and (R_3) with (R_3)−24*(R_2)

2⋅det(5)

8. Expand the 3×3 determinant using the first column.

2⋅(5*(6⋅58−(−5)⋅(−63))−(−8)*(17⋅58−(−16)⋅(−63))+1*(17⋅(−5)−(−16)⋅6))

9. Calculate the values inside the expansion.

2⋅(5*(348−315)+8*(986−1008)+1*(−85+96))

2⋅(5*(33)+8*(−22)+11)

2⋅(165−176+11)=2⋅(0)=0

Final Answer

det(1)=0

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