Inverse matrices Problem set B Answers

1. (a) \(\begin{pmatrix}3 & -2 \\ -4 & 3\end{pmatrix}\)
2. (b) \(\begin{pmatrix}1.5 & -2.5 \\ -1 & 2\end{pmatrix}\)
3. (c) No inverse: the determinant is 0.
4. (d) \(\begin{pmatrix}-2 & -1.5 \\ 1 & 0.5\end{pmatrix}\)
5. (e) \(\begin{pmatrix}1.5 & 1 \\ -0.5 & 0\end{pmatrix}\)

1. (a) \(B = A^{-1}C = \begin{pmatrix}4 & 2 \\ -11 & -2\end{pmatrix}\)
2. (b) \(B = CA^{-1} = \begin{pmatrix}-18 & 11 \\ -34 & 20\end{pmatrix}\)

\(\begin{pmatrix}0.5 & 0.5 & -0.5 \\ 0.5 & -0.5 & 0.5 \\ -0.5 & 0.5 & 0.5\end{pmatrix}\)

\(k = 2\) or \(k = -4\). Inverse: \(\dfrac{1}{k^2 + 2k - 8}\begin{pmatrix}k+2 & -4 \\ -2 & k\end{pmatrix}\).

\(\begin{pmatrix}2 & -1 \\ -5 & 3\end{pmatrix}\begin{pmatrix}59 & 11 \\ 105 & 21\end{pmatrix} = \begin{pmatrix}13 & 1 \\ 20 & 8\end{pmatrix}\), which is M, A, T, H.

1. (a) \(\begin{pmatrix}69 & 17 \\ 118 & 30\end{pmatrix}\)
2. (b) \(K^{-1} = \begin{pmatrix}2 & -1 \\ -5 & 3\end{pmatrix}\)
3. (c) SAIL

The determinant is \(10 + 3c^2\), which is at least 10, so it is never zero.

\(Q = P\begin{pmatrix}4 & 0 \\ 0 & -2\end{pmatrix}P^{-1} = \begin{pmatrix}16 & -6 \\ 36 & -14\end{pmatrix}\)

\((PQ)(Q^{-1}P^{-1}) = P(QQ^{-1})P^{-1} = PIP^{-1} = PP^{-1} = I\). The inverses come in the opposite order.