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

Problem

det(−3)

Solution

1. Identify the matrix and choose a row or column for expansion. We will use the first row to calculate the determinant.

2. Apply the cofactor expansion formula along the first row. The determinant is calculated as (a_11)*(C_11)+(a_12)*(C_12)+(a_13)*(C_13)

3. Set up the minors for each element in the first row.

det(A)=−3*det(0)−3*det(4)+3*det(4)

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

det(0)=(0)*(3)−(4)*(2)=−8

det(4)=(4)*(3)−(4)*(1)=8

det(4)=(4)*(2)−(0)*(1)=8

5. Substitute the values back into the expansion expression.

det(A)=−3*(−8)−3*(8)+3*(8)

6. Simplify the arithmetic to find the final result.

det(A)=24−24+24=24

Final Answer

det(−3)=24

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