Find the Determinant [[10,12],[-8,-10]]^15

Problem

det([[10,12],[−8,−10]]15)

Solution

1. Identify the property of determinants for powers of matrices, which states that det(An)=(det(A))n

2. Calculate the determinant of the base matrix A=[[10,12],[−8,−10]] using the formula det(A)=a*d−b*c

3. Substitute the values into the formula:

det(A)=(10)*(−10)−(12)*(−8)

4. Simplify the arithmetic expression:

det(A)=−100−(−96)

det(A)=−100+96

det(A)=−4

5. Apply the power property by raising the determinant of the base matrix to the power of 15

det(A15)=(−4)15

6. Evaluate the sign and expression. Since the exponent is odd, the result will be negative:

det(A15)=−4

Final Answer

det([[10,12],[−8,−10]]15)=−1073741824

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