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Magazine Problem 46
Use a rotation matrix to rotate the vector \(\left[\begin{array}{r}1 \\\ -2\end{array}\right]\) clockwise by the angle \(\pi / 3\)
Problem 47
$$ A=\left[\begin{array}{rr} -1 & 1 \\ 2 & 3 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & -2 \\ 3 & 2 \end{array}\right] $$ Show that \(\left(A^{-1}\right)^{-1}=A\)
Problem 47
In Problems \(47-50\), 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. $$ (-1,2) \text { and }(3,4) $$
Problem 47
Use a rotation matrix to rotate the vector \(\left[\begin{array}{r}5 \\\ -3\end{array}\right]\) clockwise by the angle \(\pi / 7\).
Problem 48
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. $$ (2,1) \text { and }(3,5) $$
Problem 48
$$ A=\left[\begin{array}{rr} -1 & 1 \\ 2 & 3 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & -2 \\ 3 & 2 \end{array}\right] $$ Show that \((A B)^{-1}=B^{-1} A^{-1}\).
Problem 49
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. \((1 .-3)\) and \((4,0)\)
Problem 49
$$ A=\left[\begin{array}{rr} -1 & 1 \\ 2 & 3 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & -2 \\ 3 & 2 \end{array}\right] $$ Find the inverse (if it exists) of $$ C=\left[\begin{array}{ll} 2 & 4 \\ 3 & 6 \end{array}\right] $$
Problem 49
In Problems 49-56, 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 \(A \mathbf{v}_{2}\). $$ A=\left[\begin{array}{rr} 2 & 3 \\ 0 & -1 \end{array}\right] $$
Problem 50
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 \(A \mathbf{v}_{2}\). $$ A=\left[\begin{array}{rr} 0 & 0 \\ 1 & -3 \end{array}\right] $$
