91Ó°ÊÓ

Prove that if \(\|A\|<1\), then \(\left\|(I+A)^{-1}\right\| \leq(1-\|A\|)^{-1}\).

Prove that if \(A\) is positive definite, then so are \(A^{2}, A^{3}, \ldots\) as well as \(A^{-1}, A^{-2}, \ldots\)

Is the inequality \(\rho(A B) \leq \rho(A) \rho(B)\) true for all pairs of \(n \times n\) matrices? Is your answer the same when \(A\) and \(B\) are upper triangular?

Solve the system $$ \left\\{\begin{array}{rr} 0.2641 x_{1}+0.1735 x_{2}+0.8642 x_{3}= & -0.7521 \\ 0.9411 x_{1}+0.0175 x_{2}+0.1463 x_{3}= & 0.6310 \\ -0.8641 x_{1}-0.4243 x_{2}+0.0711 x_{3}= & 0.2501 \end{array}\right. $$ using Gaussian elimination with: a. no pivoting b. scaled row pivoting