91Ó°ÊÓ

Use properties of the inverse to prove the given statement. If \(A\) is an \(n \times n\) invertible symmetric matrix, then \(A^{-1}\) is symmetric.

Determine the derivative of the given matrix function. $$A(t)=\left[\begin{array}{ccc} e^{t} & e^{2 t} & t^{2} \\ 2 e^{t} & 4 e^{2 t} & 5 t^{2} \end{array}\right]$$

Integration of matrix functions given in the text was done with definite integrals, but one can naturally compute indefinite integrals of matrix functions as well, by performing indefinite integrals for each element of the matrix function. For each element of the matrix \(\int A(t) d t,\) an arbitrary constant of integration must be included, and the arbitrary constants for different elements should be different. Evaluate the indefinite integral \(\int A(t) d t\) for the given matrix function. You may assume that the constants of all indefinite integrations are zero. $$A(t)=\left[\begin{array}{c} 2 t \\ 3 t^{2} \end{array}\right]$$

Consider the \(n \times n\) Hilbert matrix $$ H_{n}=\left[\frac{1}{i+j-1}\right], \quad 1 \leq i, j \leq n $$ (a) Determine \(H_{4}\) and show that it is invertible. (b) Find \(H_{4}^{-1}\) and use it to solve \(H_{4} \mathbf{x}=\mathbf{b}\) if \(\mathbf{b}=\) \([2,-1,3,5]^{T}\)