Inverting a 2x2 Matrix
Inverting a 2x2 matrix
- We earlier saw that the inverse of a matrix \(\mathbf{M}\), written as \(\mathbf{M^{-1}}\) undoes the effect of a matrix.
- This means that \(\mathbf{MM^{-1}}=\mathbf{M^{-1}M}=\mathbf{I}\)
- This is because multiplying something by a matrix followed by multiplying by the inverse of that matrix, means the original point stays the same, which is the same as multiplying by the identity matrix.
How to invert a 2x2 matrix
- If \(\mathbf{A}=\begin{pmatrix}a & b \\ c & d\end{pmatrix}\), then \(\mathbf{A^{-1}}=\frac{1}{\det(A)}\begin{pmatrix}d & -b \\ -c & a\end{pmatrix}\)
- \(\mathbf{A^{-1}}\) is the inverse of \(\mathbf{A}\), such that if \(\mathbf{A}x=y\), \(\mathbf{A^{-1}}y=x\)
- \(\mathbf{AA^{-1}}=\mathbf{A^{-1}A}=\mathbf{I}\)
- det\((\mathbf{A})=|\mathbf{A}|=ad-bc\)
Exercises on inverting a 2x2 matrix
- \(\mathbf{A}=\begin{pmatrix}2 & 0 \\ 0 & 2\end{pmatrix}\)
\(\mathbf{A^{-1}}=\frac{1}{4}\begin{pmatrix}2 & -0 \\ -0 & 2\end{pmatrix}\)
\(\mathbf{A^{-1}}=\begin{pmatrix}\frac{1}{2} & 0 \\0 & \frac{1}{2}\end{pmatrix}\)
- \(\mathbf{B}=\begin{pmatrix}1 & 2 \\3 & 4\end{pmatrix}\)
\(\mathbf{B^{-1}}=\frac{1}{4-6}\begin{pmatrix}-4 & 2 \\ 3 & -1\end{pmatrix}\)
\(\mathbf{B^{-1}}=\frac{1}{2}\begin{pmatrix}-4 & 2 \\3 & -1\end{pmatrix}\)
Matrix Proofs involving inverse
- If \(\mathbf{P}\) and \(\mathbf{Q}\) are non-singular matricies, prove that \((\mathbf{PQ})^{-1}=\mathbf{Q^{-1}P^{-1}}\)
- \(\mathbf{(PQ)^{-1}}=\mathbf{C}\)
- \((\mathbf{PQ})\mathbf{C}=\mathbf{I}\)
- \(\mathbf{P^{-1}PQC=P^{-1}I}\)
- \(\mathbf{IQC=P^{-1}}\)
- \(\mathbf{QC=P^{-1}}\)
- \(\mathbf{IC=Q^{-1}P^{-1}}\)
- \(\mathbf{C=Q^{-1}P^{-1}}\)
- \(\mathbf{(QP)^{-1}=Q^{-1}P^{-1}}\)
- If \(\mathbf{A}\) and \(\mathbf{B}\) are non-singular matricies such that \(\mathbf{BAB=I}\), prove that \(\mathbf{A=B^{-1}B^{-1}}\)
- \(\mathbf{BAB=I}\)
- \(\mathbf{B^{-1}BAB=B^{-1}I}\)
- \(\mathbf{IAB=B^{-1}}\)
- \(AB=B^{-1}\)
- \(\mathbf{ABB^{-1}=B^{-1}B^{-1}}\)
- \(\mathbf{AI=B^{-1}B^{-1}}\)
- \(\mathbf{A=B^{-1}B^{-1}}\)