## Section4.3The Inverse of a Matrix (MX3)

### Subsection4.3.1Class Activities

#### Activity4.3.1.

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 $$\RREF(A)=I$$

#### Activity4.3.2.

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 $$T\text{.}$$

#### Definition4.3.3.

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.

#### Activity4.3.4.

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 $$T^{-1}\text{.}$$

#### Observation4.3.5.

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*}

#### Activity4.3.6.

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

#### Activity4.3.7.

Is the matrix $$\left[\begin{array}{ccc} 2 & 3 & 1 \\ -1 & -4 & 2 \\ 0 & -5 & 5 \end{array}\right]$$ invertible? Give a reason for your answer.

#### Observation4.3.8.

An $$n\times n$$ matrix $$A$$ is invertible if and only if $$\RREF(A) = I_n\text{.}$$

#### Activity4.3.9.

Let $$T:\IR^2\to\IR^2$$ be the bijective linear map defined by $$T\left(\left[\begin{array}{c}x\\y\end{array}\right]\right)=\left[\begin{array}{c} 2x -3y \\ -3x + 5y\end{array}\right]\text{,}$$ with the inverse map $$T^{-1}\left(\left[\begin{array}{c}x\\y\end{array}\right]\right)=\left[\begin{array}{c} 5x+ 3y \\ 3x + 2y\end{array}\right]\text{.}$$

##### (a)

Compute $$(T^{-1}\circ T)\left(\left[\begin{array}{c}-2\\1\end{array}\right]\right)\text{.}$$

##### (b)

If $$A$$ is the standard matrix for $$T$$ and $$A^{-1}$$ is the standard matrix for $$T^{-1}\text{,}$$ find the $$2\times 2$$ matrix

\begin{equation*} A^{-1}A=\left[\begin{array}{ccc}\unknown&\unknown\\\unknown&\unknown\end{array}\right]. \end{equation*}

#### Observation4.3.10.

$$T^{-1}\circ T=T\circ T^{-1}$$ is the identity map for any bijective linear transformation $$T\text{.}$$ Therefore $$A^{-1}A=AA^{-1}$$ equals the identity matrix $$I$$ for any invertible matrix $$A\text{.}$$

### Subsection4.3.3Slideshow

Slideshow of activities available at https://teambasedinquirylearning.github.io/linear-algebra/2022/MX3.slides.html.

### Subsection4.3.5Mathematical Writing Explorations

#### Exploration4.3.11.

Assume $$A$$ is an $$n \times n$$ matrix. Prove the following are equivalent. Some of these results you have proven previously.
• $$A$$ is non-singular.

• $$A$$ row reduces to the identity matrix.

• For any choice of $$\vec{b} \in \mathbb{R}^n\text{,}$$ the system of equations represented by the augmented matrix $$[A|\vec{b}]$$ has a unique solution.

• The columns of $$A$$ are a linearly independent set.

• The columns of $$A$$ form a basis for $$\mathbb{R}^n\text{.}$$

• The rank of $$A$$ is $$n\text{.}$$

• The nullity of $$A$$ is 0.

• $$A$$ is invertible.

• The linear transformation $$T$$ with standard matrix $$A$$ is injective and surjective. Such a map is called an isomorphism.

#### Exploration4.3.12.

• Assume $$T$$ is a square matrix, and $$T^4$$ is the zero matrix. Prove that $$(I - T)^{-1} = I + T + T^2 + T^3.$$ You will need to first prove a lemma that matrix multiplication distributes over matrix addition.

• Generalize your result to the case where $$T^n$$ is the zero matrix.

### Subsection4.3.6Sample Problem and Solution

Sample problem Example B.1.20.