2.3 The Invertible Matrix Theorem

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Section 2.3 The Invertible Matrix Theorem

Handout 2.3 The Invertible Matrix Theorem

A synonym is a word you use when you can’t spell the other one.
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―
Baltasar Gracián
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The theorem introduced in this section provides many equivalent ways of describing the same idea. Depending on the context, some formulations will be more convenient than others.

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Objectives: Topics

The Invertible Matrix Theorem which is a synthesis of previously introduced concepts.

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Objectives: Goals

Characterize the invertibility of a matrix using the Invertible Matrix Theorem.

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Construct and give examples of matrices that are or are not invertible.

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Motivating Question: When is a square matrix invertible? Let me count the ways!

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Invertible matrices enjoy a rich set of equivalent descriptions.

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Theorem 2.26. The Invertible Matrix Theorem (IMT).

Let $A$ be an $n\times n$ matrix. The following statements are equivalent.

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$A$ is invertible.

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$A$ is row equivalent to $I_n\text{.}$

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$A$ has $n$ pivotal columns.

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$A\vec x=\vec 0$ has only the trivial solution.

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The columns of $A$ are linearly independent.

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The linear transformation $\vec x \mapsto A\vec x$ is one-to-one.

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The equation $A\vec x=\vec b$ has a solution for all $\vec b\in\mathbb{R}^n\text{.}$

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The columns of $A$ span $\mathbb{R}^n\text{.}$

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The linear transformation $\vec x \mapsto A\vec x$ is onto.

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There is an $n\times n$ matrix $C$ such that $CA=I_n\text{.}$ ($A$ has a left inverse.)

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There is an $n\times n$ matrix $D$ such that $AD=I_n\text{.}$ ($A$ has a right inverse.)

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$A^T$ is invertible.

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The diagram below provides another perspective on the role of the inverse matrix.

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$Diagram showing a transformation undone by its inverse.$

Diagram shows $\vec x\text{,}$ transformed by multiplication by $A\text{,}$ and reversed by multiplication by $A^{-1}\text{.}$

The matrix inverse $A^{-1}$ transforms $A\vec x$ back to $\vec x\text{,}$ because

\begin{equation*} A^{-1}(A\vec x)=(A^{-1}A)\vec x=\vec x. \end{equation*}

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Some comments on IMT:

The Invertible Matrix Theorem divides all square matrices into two classes: invertible and non-invertible.

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As the course progresses, additional equivalent conditions involving determinants, eigenvalues, and other concepts will be added.

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Note that items (j) and (k) of the Invertible Matrix Theorem lead directly to the following result.

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Theorem 2.27. Left and Right Inverses.

If $A$ and $B$ are $n\times n$ matrices and $AB=I\text{,}$ then both matrices are invertible, with $B=A^{-1}$ and $A=B^{-1}\text{.}$

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Example 2.28.

Is $A$ invertible?

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\begin{equation*} A = \begin{bmatrix} 1 \amp 0 \amp -2\\ 3 \amp 1 \amp -2\\ -5 \amp -1 \amp 9 \end{bmatrix} \end{equation*}

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Example 2.29.

For each matrix below, determine all values of $h$ so that the matrix is singular (not invertible). If there are no values of $h \text{,}$ explain why.

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