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All the four entries of the 2×2 matrix P = [[p11, p12],[p21, p22]] are nonzero, and one of its eigenvalues is zero. Which of the following statements is true?

1. p11p22 − p12p21 = 1
2. p11p22 − p12p21 = −1
3. p11p22 − p12p21 = 0
4. p11p22 + p12p21 = 0

Correct answer: p11p22 − p12p21 = 0

Solution

The determinant of a matrix is equal to the product of its eigenvalues. Since one of the eigenvalues is zero, the determinant must also be zero, which means that the expression p11p22 - p12p21, representing the determinant of the matrix, equals zero.

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