Question #59844

A student is asked to calculate the variance of the eigenvalues of a $3 \times 3$ singular matrix $A$. Due to a reading error in a single entry, the student instead uses a non-singular matrix $B$. This error alters the determinant but leaves the trace and the sum of the principal minors of order $2$ unchanged. The student correctly calculates the mean and variance of the eigenvalues of $B$ to be $4$ and $\frac{32}{3}$ respectively.

Upon discovering the error, the student recalculates the variance for the true matrix $A$. However, seeing that $A$ is singular, the student mistakenly discards the zero eigenvalue and calculates the variance of only the non-zero eigenvalues, obtaining a value $V_{\text{wrong}}$. If $V_{\text{true}}$ is the actual variance of all the eigenvalues of $A$, find the value of $|V_{\text{true}} - V_{\text{wrong}}|$.