Find the Determinant [[i,j,k],[3,-1,8],[1,1,-4]]

Problem

det(i)

Solution

1. Expand along the first row by multiplying each element in the top row by the determinant of its corresponding 2×2 minor matrix.

2. Apply the checkerboard pattern for signs, which follows the sequence + − +

3. Calculate the first minor for the i component by removing the first row and first column.

det(−1)=(−1)*(−4)−(8)*(1)=4−8=−4

4. Calculate the second minor for the j component by removing the first row and second column, remembering to negate the result.

−det(3)=−((3)*(−4)−(8)*(1))=−(−12−8)=20

5. Calculate the third minor for the k component by removing the first row and third column.

det(3)=(3)*(1)−(−1)*(1)=3+1=4

6. Combine the components to form the final vector expression.

Final Answer

det(i)=−4*i+20*j+4*k

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