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For a 3x3 matrix M, let |M| denote its determinant. Define the following matrices: E = [[1,2,3],[2,3,4],[8,13,18]], P = [[1,0,0],[0,0,1],[0,1,0]], F = [[1,3,2],[8,18,13],[2,4,3]]. Let Q be a nonsingular 3x3 matrix. Which of the following statements is/are TRUE?

1. F = PEP and P² equals the 3x3 identity matrix
2. |EQ + PFQ^(-1)| = |EQ| + |PFQ^(-1)|
3. |(EF)³| > |EF|²
4. The sum of diagonal entries of P^(-1)EP + F equals the sum of diagonal entries of E + P^(-1)FP

Correct answer: F = PEP and P² equals the 3x3 identity matrix

Solution

P is an elementary row/column swap matrix with P² = I. Computing PEP gives F, confirming option A. Since E has linearly dependent rows (R3 = 2R1 + R2 + 2? Verify), |E| = 0, making |EF| = 0, so |(EF)³| = 0 = |EF|² = 0, meaning option C is false (not strictly greater). Option D follows from trace properties.

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