Find the Determinant [[6,7,9],[7,9,9],[4,7,0]]

Problem

det(6)

Solution

1. Identify the matrix and choose a row or column for expansion. We will expand along the third row because it contains a zero.

2. Apply the cofactor expansion formula along the third row.

det(A)=(a_31)*(C_31)+(a_32)*(C_32)+(a_33)*(C_33)

3. Calculate the first term using the element 4 and its corresponding 2×2 minor.

4⋅det(7)=4*(7⋅9−9⋅9)

4*(63−81)=4*(−18)=−72

4. Calculate the second term using the element 7 Note the sign change for position (3,2)

−7⋅det(6)=−7*(6⋅9−9⋅7)

−7*(54−63)=−7*(−9)=63

5. Calculate the third term using the element 0

0⋅det(6)=0

6. Sum the results to find the final determinant.

−72+63+0=−9

Final Answer

det(6)=−9

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