Determining Possible Determinant Values for M³ = M

We are given a square matrix M such that \(M^3 = M\). We need to find the number of possible values for its determinant, denoted as \(|M|\).

Using Determinant Properties

1. Start with the given equation: \(M^3 = M\).
2. Take the determinant of both sides: \(|M^3| = |M|\).
3. Apply the determinant property \(|A^n| = |A|^n\): \(|M|^3 = |M|\).
4. Let \(x = |M|\). The equation becomes \(x^3 = x\).
5. Rearrange and solve for \(x\): \(x^3 - x = 0\) \(x(x^2 - 1) = 0\) \(x(x - 1)(x + 1) = 0\)
6. The possible values for \(x\) are \(x = 0\), \(x = 1\), and \(x = -1\).

Counting the Possible Values

The possible values for the determinant \(|M|\) are 0, 1, and -1. These are three distinct values. These values are achievable for specific matrices (e.g., the zero matrix for \(|M|=0\), the identity matrix for \(|M|=1\), and certain matrices with negative eigenvalues for \(|M|=-1\), provided the matrix dimension is odd).

Therefore, there are 3 possible values for \(|M|\).