Find the Determinant [[a,b,c],[5,6,5],[4,3,2]]

Problem

det(a)

Solution

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

2. Calculate the first minor by multiplying a by the determinant of the matrix remaining after removing the first row and first column.

det(6)=(6)*(2)−(5)*(3)=12−15=−3

3. Calculate the second minor by multiplying −b by the determinant of the matrix remaining after removing the first row and second column.

det(5)=(5)*(2)−(5)*(4)=10−20=−10

4. Calculate the third minor by multiplying c by the determinant of the matrix remaining after removing the first row and third column.

det(5)=(5)*(3)−(6)*(4)=15−24=−9

5. Combine the terms to find the final expression for the determinant.

a*(−3)−b*(−10)+c*(−9)

6. Simplify the signs and coefficients.

−3*a+10*b−9*c

Final Answer

det(a)=−3*a+10*b−9*c

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