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

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 for a 3×3 matrix along the first row.

(a_11)*((a_22)*(a_33)−(a_23)*(a_32))−(a_12)*((a_21)*(a_33)−(a_23)*(a_31))+(a_13)*((a_21)*(a_32)−(a_22)*(a_31))

3. Substitute the values from the matrix into the expansion formula.

3*(0⋅4−8⋅3)−1*(3⋅4−8⋅3)+5*(3⋅3−0⋅3)

4. Simplify the expressions inside the parentheses.

3*(0−24)−1*(12−24)+5*(9−0)

5. Calculate the products for each term.

3*(−24)−1*(−12)+5*(9)

−72+12+45

6. Sum the results to find the final determinant value.

−72+57=−15

Final Answer

det(3)=−15

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