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Magazine Problem 51
Find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) and corresponding eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\) for each matrix A. Determine the equations of the lines through the origin in the direction of the eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\), and graph the lines together with the eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\) and the vectors \(\mathrm{Av}_{1}\) and \(\mathrm{Av}_{2}\) $$A=\left[\begin{array}{rr}1 & 0 \\ 0 & -1\end{array}\right]$$
Problem 51
Parameterize the equation of the line given in standard form. $$3 x+2 y-1=0$$
Problem 52
Find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) and corresponding eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\) for each matrix A. Determine the equations of the lines through the origin in the direction of the eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\), and graph the lines together with the eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\) and the vectors \(\mathrm{Av}_{1}\) and \(\mathrm{Av}_{2}\) $$A=\left[\begin{array}{rr}-1 & 0 \\ 0 & 2\end{array}\right]$$
Problem 52
Parameterize the equation of the line given in standard form. $$x-2 y+5=0$$
Problem 52
(a) Show that if \(X=A X+D\), then $$X=(I-A)^{-1} D$$ provided that \(I-A\) is invertible. (b) Suppose that $$A=\left[\begin{array}{rr} 3 & 2 \\ 0 & -1 \end{array}\right] \text { and } D=\left[\begin{array}{r} -2 \\ 2 \end{array}\right]$$ Compute \((I-A)^{-1}\), and use your result in (a) to compute \(X\).
Problem 53
Parameterize the equation of the line given in standard form. $$2 x+y-3=0$$
Problem 53
Use the determinant to determine whether the matrix $$A=\left[\begin{array}{rr} 2 & -1 \\ -1 & 3 \end{array}\right] $$ is invertible.
Problem 53
Find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) and corresponding eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\) for each matrix A. Determine the equations of the lines through the origin in the direction of the eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\), and graph the lines together with the eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\) and the vectors \(\mathrm{Av}_{1}\) and \(\mathrm{Av}_{2}\) $$A=\left[\begin{array}{rr}-1 & 2 \\ 4 & 1\end{array}\right]$$
Problem 54
Find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) and corresponding eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\) for each matrix A. Determine the equations of the lines through the origin in the direction of the eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\), and graph the lines together with the eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\) and the vectors \(\mathrm{Av}_{1}\) and \(\mathrm{Av}_{2}\) $$A=\left[\begin{array}{rr}-1 & 0 \\ 4 & 3\end{array}\right]$$
Problem 54
Parameterize the equation of the line given in standard form. $$2 x-y+4=0$$
