Sylvester's determinant identity states that if $A$ and $B$ are matrices of sizes $m\times n$ and $n\times m$, then $$ \det(I+AB) = \det(I+BA), $$ where in the first case $I$ denotes the $m\times m$ identity, and in the second, the $n\times n$ identity. Could you sketch a proof for me, or point ...

Suppose that $x \in \mathbb R^n \,\, (n \ge 2)$ is fixed and define a matrix $A = (a_{ij})_{i,j=1}^n$ by $$a_{ij} = (1+\lvert x \rvert^2)\delta_{ij} - x_i x_j$$ where $\lvert x \rvert$ denotes the standard Euclidean norm and $\delta_{ij}$ is the Kronecker delta (i.e., $\delta_{ij} = 1$ if $i = j$...

How to find the determinant of the matrix of order $n$ with $(i,j)$th entry as $$a_{i,j}=2\delta_{i,j}-\delta_{i+1,j}-\delta_{i,j+1}$$ here $\delta_{i,j}=1$ if $i=j$ and zero otherwise? I tried it for $2\times 2$ and $3\times 3$ matrices and conclude determinant as $n+1.$ But i like directly gen...

Let $A\in M_3(\Bbb R)$ which is not a diagonal matrix. Let $P$ be a polynomial (in one variable), with real coefficients and of degree 3 such that $P(A) = 0$. Pick out the true statements: a. $P = cP_A$ where $c \in\Bbb R$ and $P_A$ is the characteristic polynomial of $A$; b. if $P$ has a compl...