Let \( A_N \) be an \( N \times N \) matrix whose entries are i.i.d. Bernoulli random variables with probability \( 1/2 \), i.e.,
\[\mathbb{P}( (A_N)_{ij} =0) = \mathbb{P}( (A_N)_{ij} =1) = \frac{1}{2}.\]
Let \( p_N \) be the probability that \( \det A_N \) is odd. Find \( \lim_{N \to \infty} p_N \).
Let \(A\), \(B\) be matrices over the reals with \(n\) rows. Let \(M=\begin{pmatrix}A &B\end{pmatrix}\). Prove that \[ \det(M^TM)\le \det(A^TA)\det(B^TB).\]
Let \( A=(a_{ij})_{ij}\) be an \(n\times n\) matrix, where \[ a_{ij}=\begin{cases} 3 &\text{if }i=j,\\ (-1)^{\lvert i-j\rvert}&\text{otherwise.}\end{cases}\] Compute the determinant of \(A\).
Let \(T\) be a tree on \(n\) vertices \(V=\{1,2,\ldots,n\}\). For two vertices \(i\) and \(j\), let \(d_{ij}\) be the distance between \(i\) and \(j\), that is the number of edges in the unique path from \(i\) to \(j\). Let \(D_T(x)=(x^{d_{ij}})_{i,j\in V}\) be the \(n\times n\) matrix. Prove that \[ \det (D_T(x))=(1-x^2)^{n-1}.\]
What is the determinant of the \(n\times n\) matrix \(A_n=(a_{ij}) \) where \[ a_{ij}=\begin{cases} 1 ,&\text{if } i=j, \\ x, &\text{if }|i-j|=1, \\ 0, &\text{otherwise,}\end{cases}\] for a real number \(x\)?
Let \(n\), \(k\) be positive integers and let \(A_1,A_2,\ldots,A_n\) be \(k\times k\) real matrices. Prove or disprove that \[ \det\left(\sum_{i=1}^n A_i^t A_i\right)\ge 0.\] (Here, \(A^t\) denotes the transpose of the matrix \(A\).)
Consider all non-empty subsets \(S_1,S_2,\ldots,S_{2^n-1}\) of \(\{1,2,3,\ldots,n\}\). Let \(A=(a_{ij})\) be a \((2^n-1)\times(2^n-1)\) matrix such that \[a_{ij}=\begin{cases}1 & \text{if }S_i\cap S_j\ne \emptyset,\\0&\text{otherwise.}\end{cases}\] What is \(\lvert\det A\rvert\)?
(This is the last problem of this semester. Good luck with your final exam!)
Let \(n\) be a fixed positive integer and let \(p\in (0,1)\). Let \(D_n\) be the determinant of a random \(n\times n\) 0-1 matrix whose entries are independent identical random variables, each of which is 1 with the probability \(p\) and 0 with the probability \(1-p\). Find the expected value and variance of \(D_n\).
Let M=(mi,j)1≤i,j≤n be an n×n matrix such that mi,j=i(i+1)(i+2)…(i+j-2). (Note that m1,1=1.) What is the determinant of M?