Let A=[a_ij] be a 22 matrix such that a_ij0,1 for all i,j.

Question

Let $A=[a_{ij}]$ be a $2	imes2$ matrix such that $a_{ij}\in\{0,1\}$ for all $i,j$. Let the random variable $X$ denote the possible values of the determinant of $A$. Then the variance of $X$ is:

Accepted Answer

$\det(A)=a_{11}a_{22}-a_{21}a_{12}\in\{-1,0,1\}$ (each entry independently $0$ or $1$ with probability $	frac12$, over $2^4=16$ equally likely matrices).

Counting: $\det=1$ for $3$ matrices ($P=3/16$), $\det=-1$ for $3$ matrices ($P=3/16$), $\det=0$ for the remaining $10$ matrices ($P=10/16$).

$E[X]=\dfrac{3}{16}(1)+\dfrac3{16}(-1)+\dfrac{10}{16}(0)=0$.

$E[X^2]=\dfrac3{16}(1)+\dfrac3{16}(1)+0=\dfrac6{16}=\dfrac38$.

$\operatorname{Var}(X)=E[X^2]-(E[X])^2=\dfrac38-0=\dfrac38$.

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