Find the Determinant [[2,4,0],[4,6,3],[-6,-10,0]]

Problem

det(2)

Solution

1. Choose a row or column for expansion. The third column is the most efficient choice because it contains two zeros.

2. Apply the cofactor expansion along the third column. The formula for the determinant using the third column is (a_13)*(C_13)+(a_23)*(C_23)+(a_33)*(C_33)

3. Identify the non-zero term in the expansion. Since (a_13)=0 and (a_33)=0 only the middle term (a_23)*(C_23) needs to be calculated.

4. Determine the sign of the cofactor (C_23) The sign is given by (−1)(2+3)=(−1)5=−1

5. Calculate the minor (M_23) by removing the second row and third column from the original matrix.

(M_23)=det(2)

6. Evaluate the 2×2 determinant using the formula a*d−b*c

(M_23)=(2)*(−10)−(4)*(−6)

(M_23)=−20−(−24)

(M_23)=4

7. Multiply the elements to find the final determinant value.

det(A)=(a_23)⋅(−1)⋅(M_23)

det(A)=3⋅(−1)⋅4

det(A)=−12

Final Answer

det(2)=−12

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