Consider an $\large n \times n$ order matrix $\large M$. The $\large i,j$-th entries of the matrix $\large M$, let's say, $\large X_{i,j}$ is an i.i.d random variable ($\large \forall i,j$) following $\large \textrm{Bernoulli }(\frac{1}{2})$. Find expectation of $\large \det(M)$, i.e. find $\large \mathbb{E}(\det(M))$, where $\large \det(M)$ implies determinant of $\large M$.
Note: $\large \det(M)$ is a random quantity.
My first approach :-
$\Large \left| \mathbf{M}^{n \times n} \right| = \begin{vmatrix} X_{11} & X_{12} & X_{13} & \cdots & X_{1n} \\ X_{21} & X_{22} & X_{23} & \cdots & X_{2n} \\ \vdots & & \ddots & \\ X_{(n-1)1} & X_{(n-1)2} & X_{(n-1)3} & \cdots & X_{(n-1)n} \\ X_{n1} & X_{n2} & X_{n3} & \cdots & X_{nn} \notag \end{vmatrix} \\ = \Large \left ( \left[\overset{n}{\underset{k=1}{\sum}} (-1)^{k+1}X_{1k} \left|\mathbf{M}^{\overline{n-1} \times \overline{n-1}} \right| \right] \right) \\ \Large \textrm{Write } \left | \mathbf{M}^{n \times n} \right| \textrm{ as } \mathbf{M_n} \textrm{ and } \left|\mathbf{M}^{\overline{n-1} \times \overline{n-1}} \right| \textrm{ as } \mathbf{M}_{1k} \\ \Large \implies \mathbf{M_n} = \left[\overset{n}{\underset{k=1}{\sum}} (-1)^{k+1}X_{1k} \mathbf{M}_{1k} \right] \\ \Large \implies \mathbb{E} \left ( \mathbf{M_n} \right) = \mathbb{E} \left ( \left[\overset{n}{\underset{k=1}{\sum}} (-1)^{k+1}X_{1k} \mathbf{M}_{1k} \right] \right) \\ \Large = \left[\overset{n}{\underset{k=1}{\sum}} (-1)^{k+1} \mathbb{E} \left( X_{1k} \mathbf{M}_{1k} \right) \right] \\ \Large = \left[\overset{n}{\underset{k=1}{\sum}} (-1)^{k+1} \mathbb{E} \left( X_{1k} \right) \mathbb{E} \left( \mathbf{M}_{1k} \right) \right] \\ \Large \textrm{The minors of } M_{1k} \text{ for } k=1,…,n \textrm{ are all } (n−1)\times(n−1) \textrm{ matrices with i.i.d. Bernoulli }\frac{1}{2} \textrm{ entries.} \\ \Large \textrm{Thus, } \mathbb{E} \left (M_{1k}\right) \textrm{ is the same for all } k.\\ \Large \therefore \textrm{Write } \mathbb{E} \left (M_{1k}\right) \textrm{ as } \mathbb{E}_{n-1} \\ \Large = \frac{1}{2} E_{n-1} \overset{n}{\underset{k=1}{\sum}} \left( -1 \right)^{k+1} \\ \Large = \underset{\text{when n =2m}}{\underbrace{0}}+ \frac{1}{2}\mathbb{E}_{n-1} \qquad [\text{when n-1 =2m}]\\ \Large \implies \mathbb{E}\left ( \mathbf{M_n} \right) = \mathbb{E}_n \qquad [ \textrm{Writing } \mathbb{E}\left ( \mathbf{M_n} \right) \textrm{ as } \mathbb{E}_n] \\ \Large = \frac{1}{2}\mathbb{E}_{n-1} \\ \Large = \frac{1}{2} \begin{cases} 1, & \textrm{ when } & n=2m+1 & m=0\\ 0, & \textrm{ when } & n= 2m+1 & m \ge 1 & [ \because \forall m\ge 1 & \mathbb{E}_{2m+1} = \frac{1}{2}. \mathbb{E}_{2m} = 0] \end{cases} $
My Second approach :-
The determinant is an alternating multilinear form. This means that swapping two rows of the matrix negates the determinant. For $\large n \ge 2$, we can use the symmetry in the distribution of the matrix $\large M$ by swapping any two rows.
Let $\large P$ be a permutation matrix that swaps any two rows. Then,
$\Large \det(PM)=−det(M) \\ \Large \implies \mathbb{E}\left[\det(PM) \right] = - \mathbb{E}\left[\det(M) \right] \qquad \longrightarrow (i)$.
Since the entries of $\large M$ are i.i.d. $\large \textsf{ Bernoulli} (\frac{1}{2})$, the distribution of $\large M$ remains unchanged under row swapping. Specifically, $\large PM$ has the same joint distribution as $\large M$ as swapping rows does not change the i.i.d. nature of the components of $\large M$.
Thus, $\Large PM \overset{\textrm{d}}{=} M \Longrightarrow \mathbb{E}[\det(PM)]=\mathbb{E}[\det(M)] \qquad \longrightarrow (ii) \\ $
Therefore from $(i)$ and $(ii)$ we have,
$\Large \mathbb{E}\left[\det(M) \right] = - \mathbb{E}\left[\det(M) \right] \\ \Large \implies 2\mathbb{E}\left[\det(M) \right] = 0 \\ \Large \therefore \mathbb{E}\left[\det(M) \right] = 0 \qquad \forall n \ge 2$
It is obvious that for $\Large n=1, \mathbb{E}\left[\det(M) \right] = \mathbb{E}\left[X_{11} \right] = \frac{1}{2}$
My questions
- Are the first and second approaches syntactically correct?
- Suppose if we want to find out the variance, then how do we calculate the variance of $\large M$ ?
Your help is highly appreciated. All comments, suggestions and answers are welcome, helpful and valuable. Thanks for help in advance.
