Find the Determinant [[1,0,1],[2,2,2],[2,2,1]]

Problem

det(1)

Solution

1. Identify the matrix and choose a row or column for cofactor expansion. Expanding along the first row is efficient because it contains a zero.

2. Apply the formula for the determinant of a 3×3 matrix using the first row elements (a_11) (a_12) and (a_13)

det(A)=(a_11)*det((a_22))−(a_12)*det((a_21))+(a_13)*det((a_21))

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

det(A)=1*det(2)−0*det(2)+1*det(2)

4. Calculate the 2×2 determinants using the rule det(a)=a*d−b*c

det(2)=(2)*(1)−(2)*(2)=2−4=−2

det(2)=(2)*(2)−(2)*(2)=4−4=0

5. Simplify the expression to find the final result.

det(A)=1*(−2)−0+1*(0)

det(A)=−2

Final Answer

det(1)=−2

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