TBIL-LA The Inverse of a Matrix (MX3)

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Section 4.3 The Inverse of a Matrix (MX3)

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Learning Outcomes

Determine if a matrix is invertible, and if so, compute its inverse.

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Subsection 4.3.1 Class Activities

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Activity 4.3.1.

Let $T : R^{n} \to R^{m}$ be a linear map with standard matrix $A .$ 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.

$A \vec{x} = \vec{b}$ has a solution for all $\vec{b} \in R^{m}$
$A \vec{x} = \vec{b}$ has a unique solution for all $\vec{b} \in R^{m}$
$A \vec{x} = \vec{0}$ has a unique solution.
The columns of $A$ span $R^{m}$
The columns of $A$ are linearly independent
The columns of $A$ are a basis of $R^{m}$
Every column of $RREF (A)$ has a pivot
Every row of $RREF (A)$ has a pivot
$m = n$ and $RREF (A) = I$

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Activity 4.3.2.

Let $T : R^{3} \to R^{3}$ be the linear transformation given by the standard matrix $A = [\begin{array}{ccc} 2 & - 1 & 0 \\ 2 & 1 & 4 \\ 1 & 1 & 3 \end{array}] .$

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

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Definition 4.3.3.

Let $T : R^{n} \to R^{n}$ be a linear map with standard matrix $A .$

If $T$ is a bijection and $\vec{b}$ is any $R^{n}$ vector, then $T (\vec{x}) = A \vec{x} = \vec{b}$ has a unique solution.
So we may define an inverse map $T^{- 1} : R^{n} \to R^{n}$ by setting $T^{- 1} (\vec{b})$ to be this unique solution.
Let $A^{- 1}$ be the standard matrix for $T^{- 1} .$ We call $A^{- 1}$ the inverse matrix of $A,$ so we also say that $A$ is invertible.

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Activity 4.3.4.

Let $T : R^{3} \to R^{3}$ be the linear transformation given by the standard matrix $A = [\begin{array}{ccc} 2 & - 1 & - 6 \\ 2 & 1 & 3 \\ 1 & 1 & 4 \end{array}] .$

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(a)

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

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(b)

Solve $T (\vec{x}) = {\vec{e}}_{2}$ to find $T^{- 1} ({\vec{e}}_{2}) .$

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(c)

Solve $T (\vec{x}) = {\vec{e}}_{3}$ to find $T^{- 1} ({\vec{e}}_{3}) .$

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(d)

Write $A^{- 1},$ the standard matrix for $T^{- 1} .$

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Observation 4.3.5.

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

[\begin{array}{cccccc} 2 & - 1 & - 6 & 1 & 0 & 0 \\ 2 & 1 & 3 & 0 & 1 & 0 \\ 1 & 1 & 4 & 0 & 0 & 1 \end{array}] \sim [\begin{array}{cccccc} 1 & 0 & 0 & 1 & - 2 & 3 \\ 0 & 1 & 0 & - 5 & 14 & - 18 \\ 0 & 0 & 1 & 1 & - 3 & 4 \end{array}]

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Activity 4.3.6.

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

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Activity 4.3.7.

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

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Observation 4.3.8.

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

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Activity 4.3.9.

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

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(a)

Compute $(T^{- 1} \circ T) ([\begin{matrix} - 2 \\ 1 \end{matrix}]) .$

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(b)

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

A^{- 1} A = [\begin{array}{ccc} ? & ? \\ ? & ? \end{array}] .

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Observation 4.3.10.

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

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Subsection 4.3.2 Videos

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Figure 44. Video: Invertible matrices

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Figure 45. Video: Finding the inverse of a matrix

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Exploration 4.3.11.

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 $\vec{b} \in R^{n},$ 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 $R^{n} .$
The rank of $A$ is $n .$
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.

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Exploration 4.3.12.

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.

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Subsection 4.3.6 Sample Problem and Solution

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Sample problem Example B.1.20.

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