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Section 4.3 The Inverse of a Matrix (M3)

Activity 4.3.1.

Let \(T: \IR^n \rightarrow \IR^m\) be a linear map with standard matrix \(A\text{.}\) Sort the following items into three groups of statements: a group that means \(T\) is injective, a group that means \(T\) is surjective, and a group that means \(T\) is bijective.

  1. \(A\vec x=\vec b\) has a solution for all \(\vec b\in\IR^m\)

  2. \(A\vec x=\vec b\) has a unique solution for all \(\vec b\in\IR^m\)

  3. \(A\vec x=\vec 0\) has a unique solution.

  4. The columns of \(A\) span \(\IR^m\)

  5. The columns of \(A\) are linearly independent

  6. The columns of \(A\) are a basis of \(\IR^m\)

  7. Every column of \(\RREF(A)\) has a pivot

  8. Every row of \(\RREF(A)\) has a pivot

  9. \(m=n\) and \(\RREF(A)=I\)

Activity 4.3.2.

Let \(T: \IR^3 \rightarrow \IR^3\) be the linear transformation given by the standard matrix \(A=\left[\begin{array}{ccc} 2 & -1 & 0 \\ 2 & 1 & 4 \\ 1 & 1 & 3 \end{array}\right]\text{.}\)

Write an augmented matrix representing the system of equations given by \(T(\vec x)=\vec{0}\text{,}\) that is, \(A\vec x=\left[\begin{array}{c}0 \\ 0 \\ 0 \end{array}\right]\text{.}\) Then solve \(T(\vec x)=\vec{0}\) to find the kernel of \(T\text{.}\)

Definition 4.3.3.

Let \(T: \IR^n \rightarrow \IR^n\) be a linear map with standard matrix \(A\text{.}\)

  • If \(T\) is a bijection and \(\vec b\) is any \(\IR^n\) vector, then \(T(\vec x)=A\vec x=\vec b\) has a unique solution.

  • So we may define an inverse map \(T^{-1} : \IR^n \rightarrow \IR^n\) by setting \(T^{-1}(\vec b)\) to be this unique solution.

  • Let \(A^{-1}\) be the standard matrix for \(T^{-1}\text{.}\) We call \(A^{-1}\) the inverse matrix of \(A\text{,}\) so we also say that \(A\) is invertible.

Activity 4.3.4.

Let \(T: \IR^3 \rightarrow \IR^3\) be the linear transformation given by the standard matrix \(A=\left[\begin{array}{ccc} 2 & -1 & -6 \\ 2 & 1 & 3 \\ 1 & 1 & 4 \end{array}\right]\text{.}\)

(a)

Write an augmented matrix representing the system of equations given by \(T(\vec x)=\vec{e}_1\text{,}\) that is, \(A\vec x=\left[\begin{array}{c}1 \\ 0 \\ 0 \end{array}\right]\text{.}\) Then solve \(T(\vec x)=\vec{e}_1\) to find \(T^{-1}(\vec{e}_1)\text{.}\)

(b)

Solve \(T(\vec x)=\vec{e}_2\) to find \(T^{-1}(\vec{e}_2)\text{.}\)

(c)

Solve \(T(\vec x)=\vec{e}_3\) to find \(T^{-1}(\vec{e}_3)\text{.}\)

(d)

Write \(A^{-1}\text{,}\) the standard matrix for \(T^{-1}\text{.}\)

Observation 4.3.5.

We could have solved these three systems simultaneously by row reducing the matrix \([A\,|\,I]\) at once.

\begin{equation*} \left[\begin{array}{ccc|ccc} 2 & -1 & -6 & 1 & 0 & 0 \\ 2 & 1 & 3 & 0 & 1 & 0 \\ 1 & 1 & 4 & 0 & 0 & 1 \end{array}\right] \sim \left[\begin{array}{ccc|ccc} 1 & 0 & 0 & 1 & -2 & 3 \\ 0 & 1 & 0 & -5 & 14 & -18 \\ 0 & 0 & 1 & 1 & -3 & 4 \end{array}\right] \end{equation*}

Activity 4.3.6.

Find the inverse \(A^{-1}\) of the matrix \(A=\left[\begin{array}{cc} 1 & 3 \\ 0 & -2 \end{array}\right]\) by row-reducing \([A\,|\,I]\text{.}\)

Subsection 4.3.1 Videos

Figure 4.3.7. Video: Invertible matrices

Exercises 4.3.2 Exercises

Exercises available at checkit.clontz.org 1 .

https://checkit.clontz.org/#/banks/tbil-la/M3/