91Ó°ÊÓ

Find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) for each matrix \(A\). $$A=\left[\begin{array}{rr}-7 & 0 \\ 0 & 6\end{array}\right]$$

Suppose that $$ A=\left[\begin{array}{ll} a & 8 \\ 2 & 4 \end{array}\right], \quad X=\left[\begin{array}{l} x \\ y \end{array}\right], \quad \text { and } \quad B=\left[\begin{array}{l} b_{1} \\ b_{2} \end{array}\right] $$ (a) Show that when \(a \neq 4, A X=B\) has exactly one solution. (b) Suppose \(a=4\). Find conditions on \(b_{1}\) and \(b_{2}\) such that \(A X=B\) has (i) infinitely many solutions and (ii) no solutions. (c) Explain your results in (a) and (b) graphically.

Find the parametric equation of the line in \(x-y-z\) space that goes through the indicated point in the direction of the indicated vector. $$(2,1,-3),\left[\begin{array}{r}3 \\ -1 \\ 2\end{array}\right]$$

Write down the inverse of \(A\). $$ A=\left[\begin{array}{rr} 2 & 1 \\ -3 & -1 \end{array}\right] $$

Find the parametric equation of the line in \(x-y-z\) space that goes through the given points. \((5,4,-1)\) and \((2,0,3)\)

Find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) for each matrix \(A\). $$A=\left[\begin{array}{lr}1 & -3 \\ 0 & 2\end{array}\right]$$

Find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) for each matrix \(A\). $$A=\left[\begin{array}{rr}-1 & 4 \\ 0 & -2\end{array}\right]$$

Find the parametric equation of the line in \(x-y-z\) space that goes through the given points. \((2,0,-3)\) and \((4,1,1)\)

Write down the inverse of \(A\). $$ A=\left[\begin{array}{ll} 1 & 2 \\ 1 & 3 \end{array}\right] $$

Find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) for $$ A=\left[\begin{array}{ll} a & 0 \\ c & b \end{array}\right] $$