Find the Determinant [[5,2,9],[0,6,9],[5,4,8]]

Problem

det(5)

Solution

1. Identify the matrix and choose a row or column for expansion. We will expand along the first column because it contains a zero, which simplifies the calculation.

2. Apply the cofactor expansion formula along the first column. The determinant is calculated as the sum of the elements multiplied by their corresponding cofactors.

det(A)=(a_11)⋅(C_11)+(a_21)⋅(C_21)+(a_31)⋅(C_31)

3. Calculate the first term using the element 5 at position (1,1)

5⋅det(6)=5⋅(6⋅8−9⋅4)

5⋅(48−36)=5⋅12=60

4. Calculate the second term using the element 0 at position (2,1)

−0⋅det(2)=0

5. Calculate the third term using the element 5 at position (3,1)

5⋅det(2)=5⋅(2⋅9−9⋅6)

5⋅(18−54)=5⋅(−36)=−180

6. Sum the results of the cofactor expansions to find the final determinant.

60+0−180=−120

Final Answer

det(5)=−120

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