Find the Determinant [[1,0,4],[4,1,9],[k,7,-4]]

Problem

det(1)

Solution

1. Set up the expansion along the first row to simplify calculations since it contains a zero.

2. Apply the formula for the determinant of a 3×3 matrix using the entries of the first row and their corresponding 2×2 minors.

1⋅det(1)−0⋅det(4)+4⋅det(4)

3. Calculate the 2×2 determinants using the rule a*d−b*c

det(1)=(1)*(−4)−(9)*(7)=−4−63=−67

det(4)=(4)*(7)−(1)*(k)=28−k

4. Substitute these values back into the expansion expression.

1*(−67)−0+4*(28−k)

5. Distribute and combine like terms to find the final expression.

−67+112−4*k

45−4*k

Final Answer

det(1)=45−4*k

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