Systems and matrices · Definition

Inverse matrix

linear algebramatrices

Before you start you need to know

The inverse matrix undoes what a matrix does: it is the analogue of the reciprocal of a number, but it exists only for some square matrices.

Definition

The inverse of a square matrix \(A\) is the matrix \(A^{-1}\) that gives the identity \(I\) when multiplied by \(A\) on either side; if it exists, \(A\) is called invertible.

\[AA^{-1}=A^{-1}A=I\]

With numbers, solving \(ax=b\) means multiplying by \(1/a\). For matrices we want something similar for \(A\mathbf{x}=\mathbf{b}\): a matrix \(A^{-1}\) which, multiplied on the left, undoes the effect of \(A\). The condition \(AA^{-1}=A^{-1}A=I\) says exactly that: applying \(A\) and then \(A^{-1}\) (or the other way round) leaves every vector where it started.

Not every matrix has an inverse, just as the number 0 has no reciprocal. If \(A\) squashes some nonzero vector down to zero, or if its rows are dependent, information is lost and nothing can recover it. That is why the \([A\,|\,I]\) method is also a test: if elimination produces a row of zeros, the matrix is not invertible.

Guiding example: for \(A=\begin{pmatrix}1&2\\3&5\end{pmatrix}\), reducing \([A\,|\,I]\) gives \(A^{-1}=\begin{pmatrix}-5&2\\3&-1\end{pmatrix}\). A typical mistake is to invert entry by entry (writing \(1/a_{ij}\)); another is to forget that \((AB)^{-1}=B^{-1}A^{-1}\), with the order reversed. In practice one rarely computes \(A^{-1}\) to solve a system, since elimination is cheaper, but the inverse is central to the theory and connects with the determinant, which is nonzero exactly when the inverse exists. When the inverse exists, it is unique.

Think about it

If \(A\) and \(B\) are invertible of the same size, what is the inverse of \(AB\)?

\(B^{-1}A^{-1}\): to undo «first \(B\), then \(A\)» you must undo \(A\) first and then \(B\).

Why can a matrix with a row of zeros not be invertible?

Because \(A\mathbf{x}\) would always have a zero in that component, so it could not produce an arbitrary \(\mathbf{b}\); there is no way to undo the transformation.

Exercises 10

  1. Compute the inverse of \(\begin{pmatrix}2 & 1 \\ 5 & 3\end{pmatrix}\).
  2. Which of these matrices has NO inverse?
  3. Compute the inverse of the diagonal matrix \(\begin{pmatrix}2 & 0 & 0 \\ 0 & -4 & 0 \\ 0 & 0 & 5\end{pmatrix}\).
  4. If \(A\) and \(B\) are invertible matrices of the same size, what is \((AB)^{-1}\)?
  5. Solve \(A\mathbf{x}=\mathbf{b}\) using the inverse, with \(A=\begin{pmatrix}1 & 2 \\ 3 & 5\end{pmatrix}\) and \(\mathbf{b}=(1,2)^T\).
  6. A square matrix \(A\) satisfies \(A^2-3A+I=0\). Which expression is \(A^{-1}\)?
  7. Let \(A=\begin{pmatrix}1 & 2 & 0 \\ 0 & 1 & 3 \\ 0 & 0 & 1\end{pmatrix}\). What is the entry in row 1, column 3 of \(A^{-1}\)?
  8. If \(A\) is an invertible \(2\times2\) matrix, what is \((3A)^{-1}\)?
  9. Select all values of \(k\) for which \(\begin{pmatrix}k & 2\\ 8 & k\end{pmatrix}\) is NOT invertible.
  10. Let \(A\) and \(B\) be square matrices of the same size with \(AB=I\). What can be stated?

References

  • Gilbert Strang, Introduction to Linear Algebra, 6th edition, Wellesley-Cambridge Press, 2023. §2.5
  • Sheldon Axler, Linear Algebra Done Right, 4th edition, Springer, 2024. §3.D