Systems of equations with the inverse matrix Problem set A Answers

1. (a) \(x = 3,\; y = 1\)
2. (b) \(x = -2.5,\; y = 4\)
3. (c) \(x = 2,\; y = -1\)

1. (a) \(x = 1,\; y = 1,\; z = 2\)
2. (b) \(x = 1,\; y = 2,\; z = 3\)

1. (a) \(k = \pm 6\)
2. (b) \(k = \pm 3\)

1. (a) 0
2. (b) k = 8
3. (c) No solutions: the lines are parallel and different.

\(\begin{pmatrix}3 & 2 \\ 4 & 5\end{pmatrix}\begin{pmatrix}c \\ t\end{pmatrix} = \begin{pmatrix}15.8 \\ 26.2\end{pmatrix}\). Coffee €3.80, tea €2.20.

\(\begin{pmatrix}1 & 1 & 1 \\ 1 & 0 & -1 \\ 0.04 & 0.025 & 0.06\end{pmatrix}\begin{pmatrix}a \\ b \\ c\end{pmatrix} = \begin{pmatrix}40000 \\ 5000 \\ 1575\end{pmatrix}\). A $15 000, B $15 000, C $10 000.

\(x = \dfrac{2k - 1}{k^2 - 1},\quad y = \dfrac{k - 2}{k^2 - 1}\)