Exercise-4

If $A$ and $B$ are two square matrices such that $B=-A^{-1}BA,$ then show that ${\left(A+B\right)}^2=A^2+B^2$.

If $A=\left[\begin{array}{lll}a& 1& 0\\ 1& b& d\\ 1& b& c\end{array}\right], B=\left[\begin{array}{lll}a& 1& 1\\ 0& d& c\\ f& g& h\end{array}\right], U=\left[\begin{array}{l}f\\ g\\ h\end{array}\right], V=\left[\begin{array}{l}{a}^2\\ 0\\ 0\end{array}\right], X=\left[\begin{array}{l}x\\ y\\ z\end{array}\right]$ and $AX=U$ has infinitely many solutions, prove that $BX=V$ has no unique solution. Also, prove that if $afd$ $\ne 0,$ then $BX=V$ has no solution.

If the system of equations $x=cy+bz, y=az+cx$ and $z=bx+ay$ has a non-zero solution and at least one of $a, b, c$ is a proper fraction, then prove that $a^2+b^2+c^2<3$ and $abc>-1$.

If a diagonal matrix $D=diag\left\{d_1,d_2,\dots \dots \dots ,d_n\right\},$ then prove that $f\left(D\right)=diag\left\{f\left(d_1\right),f\left(d_2\right),\dots \dots \dots f\left(d_n\right)\right\},$ where $f\left(x\right)$ is a polynomial with scalar coefficient.

Given the matrix $\mathrm{A}=\left[\begin{array}{ccc}-1& 3& 5\\ 1& -3& -5\\ -1& 3& 5\end{array}\right]$ and $\mathrm{X}$ be the solution set of the equation $A^x=\mathrm{A},$ where $x\in N-\left\{1\right\}$. Evaluate $\prod \frac{x^3+1}{x^3-1},$ where the continued product extends $\forall x\in X$.

If three are three square matrices $A, B, C$ of same order satisfying the equation $A^3=A^{-1}$ and let $B=-A^{3^n}$ and $C=A^{3^{\left(n+4\right)}}$, then prove that $det(B+C)=0, n\in N$

If $A$ is a non-singular matrix satisfying $AB-BA=A,$ then prove that $det\left(B+I\right)=det\left(B-I\right)$.

Without expanding the determinant. Prove that

$\left|\begin{array}{ccc}n{a}_1+b_1& n{a}_2+b_2& n{a}_3+b_3\\ n{b}_1+c_1& n{b}_2+c_2& n{b}_3+c_3\\ n{c}_1+a_1& n{c}_2+a_2& n{c}_3+a_3\end{array}\right|=\left(n^3+1\right)\left|\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|$