Linear Algebra with SageMath A Gentle Introduction

Find the eigenvalues and corresponding eigenvectors of the following matrices

Let \(A=\begin{pmatrix} -3 \amp -2\amp 1\\-2\amp 0\amp 4\\-6\amp -3\amp 5\end{pmatrix}\text{.}\) Find the eigenvalues of \(3A^3-5A^2+4I\)

Let \(A\begin{pmatrix} -3 \amp -2\amp 1\\-2\amp 0\amp 4\\-6\amp -3\amp 5\end{pmatrix}\text{.}\) Show that \(A\) satisfies its characteristic equation and hence find \(A^{-1}\text{.}\) Also find \(A^4\text{.}\)

Check if the following matrices are positive definite? \(A_1=\begin{pmatrix} 1\amp -7 \\-7\amp 3 \end{pmatrix}; A_2=\begin{pmatrix} 2 \amp 1\amp 3\\1\amp 4\amp -1\\3\amp -1\amp 2\end{pmatrix}; A_3=\begin{pmatrix} -3 \amp -2\amp 1\\-2\amp 0\amp 4\\1\amp 4\amp 5\end{pmatrix}\)

Find eigenvalues of \(A_2 = \begin{pmatrix} 0 \amp 1\\1 \amp 0\end{pmatrix}\) and \(A_3=\begin{pmatrix} 0 \amp 1\amp 1\\1\amp 0\amp 1\\1\amp 1\amp 0\end{pmatrix}\text{.}\) Can you generalize this?

Find the algebraic and geometric multiplicities of each of the eigenvalues of \(A = \begin{pmatrix} 2 \amp 0 \amp 0 \amp 0\\0 \amp 2 \amp 0 \amp 0\\0 \amp 0 \amp -1 \amp 1\\0 \amp 0 \amp 0 \amp -1\end{pmatrix}\text{.}\) Is this matrix diagonalizable?

Consider the matrix \(A=\left(\begin{array}{rr} 0 \amp \frac{1}{2} \\ -1 \amp \frac{3}{2} \end{array}\right)\text{.}\) What is \(\displaystyle\lim_{n\to \infty} A^n\text{?}\)

Let \(A\) be a \(3\times 3\) real matrix with eigenvalues \(-1,0,1\) and corresponding eigenvectors \((1, -4, 1), (1, -3, 1), (1, -2, 1/2)\) respectively. Find \(A\text{.}\)

Suppose there are two internet service providers \(A\) and \(B\) in a city. At present \(A\) has 50000 subscriber and \(B\) has 100000 subscribers. A trend shows that every year 60% of \(A\) subscriber move to \(B\) and 40% of \(B\) move to \(A\text{.}\) After 2 year how many subscribers \(A\) and \(B\) will have? What happens in long run?