91Ó°ÊÓ

Determine conditions on the \(b\) 's, if any, in order to guarantee that the linear system is consistent. $$\begin{aligned} &6 x_{1}-4 x_{2}=b_{1}\\\ &3 x_{1}-2 x_{2}=b_{2} \end{aligned}$$

Determine conditions on the \(b\) 's, if any, in order to guarantee that the linear system is consistent. $$\begin{aligned} x_{1}-2 x_{2}+5 x_{3} &=b_{1} \\ 4 x_{1}-5 x_{2}+8 x_{3} &=b_{2} \\ -3 x_{1}+3 x_{2}-3 x_{3} &=b_{3} \end{aligned}$$

Use the inversion algorithm to find the inverse of the given matrix, if the inverse exists. $$\left[\begin{array}{rrr} -1 & 3 & -4 \\ 2 & 4 & 1 \\ -4 & 2 & -9 \end{array}\right]$$

Find the cubic polynomial whose graph passes through the points \((-1,-1),(0,1),(1,3),(4,-1).\)

Use the given information to find \(A\). $$(7 A)^{-1}=\left[\begin{array}{rr} -3 & 7 \\ 1 & -2 \end{array}\right]$$

Find all values of \(k,\) if any, that satisfy the equation. $$\left[\begin{array}{lll} k & 1 & 1 \end{array}\right]\left[\begin{array}{lll} 1 & 1 & 0 \\ 1 & 0 & 2 \\ 0 & 2 & -3 \end{array}\right]\left[\begin{array}{l} k \\ 1 \\ 1 \end{array}\right]=0$$ Answer: -1

Decide whether the given matrix is symmetric. $$\left[\begin{array}{rr}0 & -7 \\ -7 & 7\end{array}\right]$$

Determine conditions on the \(b\) 's, if any, in order to guarantee that the linear system is consistent. $$\begin{aligned} x_{1}-2 x_{2}-x_{3} &=b_{1} \\ -4 x_{1}+5 x_{2}+2 x_{3} &=b_{2} \\ -4 x_{1}+7 x_{2}+4 x_{3} &=b_{3} \end{aligned}$$

Use the inversion algorithm to find the inverse of the given matrix, if the inverse exists. $$\left[\begin{array}{lll} \frac{1}{5} & \frac{1}{5} & -\frac{2}{5} \\ \frac{1}{5} & \frac{1}{5} & \frac{1}{10} \\ \frac{1}{5} & -\frac{4}{5} & \frac{1}{10} \end{array}\right]$$

Find all values of \(k,\) if any, that satisfy the equation. $$\left[\begin{array}{lll} 2 & 2 & k \end{array}\right]\left[\begin{array}{lll} 1 & 2 & 0 \\ 2 & 0 & 3 \\ 0 & 3 & 1 \end{array}\right]\left[\begin{array}{l} 2 \\ 2 \\ k \end{array}\right]=0$$