We have seen how a matrix transformation can perform a geometric operation; now we would like to find a matrix transformation that undoes that operation.

  1. Suppose that \(T:\real^2\to\real^2\) is the matrix transformation that rotates vectors by \(90^\circ\text{.}\) Find a matrix transformation \(S:\real^2\to\real^2\) that undoes the rotation; that is, \(S\) takes \(T(\xvec)\) back into \(\xvec\) so that \(S\circ T(\xvec) = \xvec\text{.}\) Think geometrically about what the transformation \(S\) should be and then verify it algebraically.

    We say that \(S\) is the inverse of \(T\) and we will write it as \(T^{-1}\text{.}\)

  2. Verify algebraically that the reflection \(R:\real^2\to\real^2\) across the line \(y=x\) is its own inverse; that is, \(R^{-1} = R\text{.}\)

  3. The matrix transformation \(T:\real^2\to\real^2\) defined by the matrix

    \begin{equation*} A=\left[\begin{array}{rr} 1\amp 1 \\ 0 \amp 1\\ \end{array}\right] \end{equation*}

    is called a shear. Find the inverse of \(T\text{.}\)

  4. Describe the geometric effect of the matrix transformation defined by

    \begin{equation*} A=\left[\begin{array}{rr} \frac12 \amp 0 \\ 0 \amp 3\\ \end{array}\right] \end{equation*}

    and then find its inverse.