Find the Determinant [[4,-6,2],[8,0,1],[3,5,4]]

Problem

det(4)

Solution

1. Choose a row or column for expansion. We will expand along the first row using the elements 4 −6 and 2

2. Apply the cofactor expansion formula for a 3×3 matrix. The determinant is calculated as (a_11)*(C_11)+(a_12)*(C_12)+(a_13)*(C_13)

3. Set up the minors by removing the row and column corresponding to each element in the first row.

det(A)=4*det(0)−(−6)*det(8)+2*det(8)

4. Calculate the 2×2 determinants using the formula a*d−b*c

det(0)=(0)*(4)−(1)*(5)=−5

det(8)=(8)*(4)−(1)*(3)=29

det(8)=(8)*(5)−(0)*(3)=40

5. Substitute the values back into the expansion expression.

det(A)=4*(−5)+6*(29)+2*(40)

6. Simplify the arithmetic to find the final value.

det(A)=−20+174+80

det(A)=234

Final Answer

det(4)=234

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