Inverse matrices Problem set A

Find the inverse of each matrix, or explain why there is none.

1. (a) \(\begin{pmatrix}2 & 1 \\ 5 & 3\end{pmatrix}\)
2. (b) \(\begin{pmatrix}4 & 3 \\ 2 & 2\end{pmatrix}\)
3. (c) \(\begin{pmatrix}6 & 9 \\ 2 & 3\end{pmatrix}\)
4. (d) \(\begin{pmatrix}1 & -2 \\ 3 & -4\end{pmatrix}\)
5. (e) \(\begin{pmatrix}0 & 2 \\ -1 & 5\end{pmatrix}\)

\(A = \begin{pmatrix}2 & 1 \\ 5 & 3\end{pmatrix}\) and \(C = \begin{pmatrix}4 & -1 \\ 9 & 0\end{pmatrix}\). Find \(B\) when:

1. (a) \(AB = C\)
2. (b) \(BA = C\)

Use your calculator to find the inverse of \(\begin{pmatrix}1 & 0 & 2 \\ 0 & 1 & 1 \\ 1 & 1 & 0\end{pmatrix}\).

Find the values of \(k\) for which \(\begin{pmatrix}k & 2 \\ 3 & k-1\end{pmatrix}\) is singular. For all other values of \(k\), write down its inverse.

A four-letter word is written as numbers (A = 1, B = 2, …) in a 2 × 2 matrix, row by row. It is put into code by multiplying on the left by \(\begin{pmatrix}2 & 1 \\ 5 & 3\end{pmatrix}\). The coded matrix is \(\begin{pmatrix}28 & 26 \\ 76 & 73\end{pmatrix}\). Find the word.

Two friends send each other messages in code. A word is turned into numbers (A = 1, B = 2, up to Z = 26), written in pairs down the columns of a matrix and multiplied on the left by the key \(K = \begin{pmatrix}2 & 1 \\ 5 & 3\end{pmatrix}\). They then send the four numbers of the product. Alma encodes the word SAND. Her friend's reply is coded with the same key as \(\begin{pmatrix}31 & 31 \\ 85 & 80\end{pmatrix}\).

1. (a) Encode the word SAND.
2. (b) Find \(K^{-1}\).
3. (c) Decode the second message.

Show that \(\begin{pmatrix}1 & 2c \\ -c & 3\end{pmatrix}\) has an inverse for every value of \(c\).

\(P = \begin{pmatrix}1 & 2 \\ 1 & 3\end{pmatrix}\). Find the matrix \(Q\) such that \(P^{-1}QP = \begin{pmatrix}3 & 0 \\ 0 & -1\end{pmatrix}\).

By working out \((AB)(B^{-1}A^{-1})\), show that \((AB)^{-1} = B^{-1}A^{-1}\).