91Ó°ÊÓ

Find the inverse (if it exists) of $$ I_{3}=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] $$

Find the parametric equation of the line in the \(x-y\) plane that goes through the given points. Then eliminate the parameter to find the equation of the line in standard form. \((2,3)\) and \((-1,-4)\)

In Problems \(51-54\), parameterize the equation of the line given in standard form. $$ 3 x+4 y-1=0 $$

In Problems , find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) and corresponding eigenvectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) for each matrix A. Determine the equations of the lines through the origin in the direction of the eigenvectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\), and graph the lines together with the eigenvectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) and the vectors \(A \mathbf{v}_{1}\) and \(A \mathbf{v}_{2}\). $$ A=\left[\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right] $$

Suppose that $$ A=\left[\begin{array}{rr} -1 & 0 \\ 2 & -3 \end{array}\right] \text { and } D=\left[\begin{array}{l} -2 \\ -5 \end{array}\right] $$ Find \(X\) such that \(A X=D\) by (a) solving the associated system of linear equations and (b) using the inverse of \(A\).

Parameterize the equation of the line given in standard form. $$ x-2 y+5=0 $$

(a) Show that if \(X=A X+D\), then $$ X=(I-A)^{-1} D $$ provided that \(I-A\) is invertible. (b) Suppose that $$ A=\left[\begin{array}{rr} 3 & 2 \\ 0 & -1 \end{array}\right] \text { and } \quad D=\left[\begin{array}{r} -2 \\ 2 \end{array}\right] $$ Compute \((I-A)^{-1}\), and use your result in (a) to compute \(X\).

In Problems , find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) and corresponding eigenvectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) for each matrix A. Determine the equations of the lines through the origin in the direction of the eigenvectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\), and graph the lines together with the eigenvectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) and the vectors \(A \mathbf{v}_{1}\) and \(A \mathbf{v}_{2}\). $$ A=\left[\begin{array}{rr} -1 & 0 \\ 0 & 2 \end{array}\right] $$

In Problems , find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) and corresponding eigenvectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) for each matrix A. Determine the equations of the lines through the origin in the direction of the eigenvectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\), and graph the lines together with the eigenvectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) and the vectors \(A \mathbf{v}_{1}\) and \(A \mathbf{v}_{2}\). $$ A=\left[\begin{array}{ll} -4 & 2 \\ -3 & 1 \end{array}\right] $$

Use the determinant to determine whether the matrix $$ A=\left[\begin{array}{rr} 2 & -1 \\ 1 & 3 \end{array}\right] $$ is invertible.