Find the Determinant [[1,4,7],[10,13,16],[19,22,25]]

Problem

det(1)

Solution

1. Apply the expansion formula along the first row to break the 3×3 determinant into 2×2 determinants.

1*det(13)−4*det(10)+7*det(10)

2. Calculate the first minor by finding the product of the diagonals.

13⋅25−16⋅22=325−352=−27

3. Calculate the second minor using the same cross-multiplication method.

10⋅25−16⋅19=250−304=−54

4. Calculate the third minor using the same cross-multiplication method.

10⋅22−13⋅19=220−247=−27

5. Combine the results by multiplying each minor by its corresponding coefficient from the first row.

1*(−27)−4*(−54)+7*(−27)

6. Simplify the expression to find the final value.

−27+216−189=0

Final Answer

det(1)=0

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