Basics of the inverse matrix逆行列ぎゃくぎょうれつ

1Introduction

The most important point in this lecture is that an inverse matrix逆行列ぎゃくぎょうれつ is not merely a formula to memorize. It should be understood as the question of whether the map represented by a matrix can be recovered.

If a matrix行列ぎょうれつ $A$A sends an input to an output, then $A^{-1}$A^{-1} recovers the original input uniquely from that output. When information is lost, no inverse matrix exists.

2Terms and definitions

An inverse matrix逆行列ぎゃくぎょうれつ of a square matrix正方行列せいほうぎょうれつ $A$A is a matrix satisfying

The essential point is that it is a two-sided inverse: multiplication from both the left and the right gives the identity matrix単位行列たんいぎょうれつ. The identity matrix is the matrix $I$I that changes nothing when multiplied.

3Strategy

First interpret an inverse matrix逆行列ぎゃくぎょうれつ as the map that recovers the input from the output. Then check that the process of using Gaussian elimination to transform $A$A into $I$I is exactly the process that constructs $A^{-1}$A^{-1}.

4Intuitive explanation

A pure expansion can be reversed by shrinking, and a pure rotation can be reversed by rotating in the opposite direction. But a map that collapses a plane onto a single line loses information, so it cannot be recovered. This is the situation in which no inverse matrix逆行列ぎゃくぎょうれつ exists.

5Rigorous explanation

5.11. Definition

Let $A$A be an $n\times n$n\times n square matrix正方行列せいほうぎょうれつ. If $A^{-1}$A^{-1} exists, then

For a rectangular matrix, the left and right sizes do not match, so an inverse matrix in this two-sided sense is not defined.

5.22. Computing by elimination

Form $(A\mid I)$(A\mid I) and use elimination to turn the left side into $I$I. Then the right side becomes $A^{-1}$A^{-1}.

5.33. When no inverse exists

If a pivot cannot be secured during elimination and the rank階数かいすう is deficient, then no inverse matrix逆行列ぎゃくぎょうれつ exists.

5.44. Equivalent criteria

For an $n\times n$n\times n matrix $A$A, the following conditions are equivalent.

The determinant criterion is a preview of the later determinant lectures. The main criteria used in this page are elimination, rank, kernel, and uniqueness of solutions.

1. $A^{-1}$A^{-1} exists.
2. $\mathrm{rank}\left(A\right)=n$\operatorname{rank}(A)=n.
3. $\ker \left(A\right)=\left\{0\right\}$\ker(A)=\{0\}.
4. $A:K^n\to K^n$A:K^n\to K^n is injective and surjective.
5. For every $b\in K^n$b\in K^n, $Ax=b$Ax=b has a unique solution.
6. $\det A\ne 0$\det A\ne0 (the determinant criterion introduced later).
7. Elimination has a pivot in every column.

This list shows that an inverse matrix is the intersection of several tests saying that no information is lost. The condition $\ker \left(A\right)=\left\{0\right\}$\ker(A)=\{0\} means different inputs do not collapse to the same output, so any existing solution of $Ax=b$Ax=b is unique. In finite dimension, rank-nullity then gives $\mathrm{rank}\left(A\right)=n$\operatorname{rank}(A)=n, so the image is all of $K^n$K^n and a solution exists for every $b$b. Sending $b$b to that unique solution defines $A^{-1}$A^{-1}.

5.55. One-sided inverses for rectangular matrices

If $A$A is an $m\times n$m\times n matrix, one-sided inverses can be considered instead. A left inverse $L$L satisfies $LA=I_n$LA=I_n and exists when $\mathrm{rank}\left(A\right)=n$\operatorname{rank}(A)=n. A right inverse $R$R satisfies $AR=I_m$AR=I_m and exists when $\mathrm{rank}\left(A\right)=m$\operatorname{rank}(A)=m.

One-sided inverses need not be unique. When columns or rows are missing, a criterion is needed to choose a least-squares solution or a minimum-norm solution. The pseudoinverse擬似逆行列ぎじぎゃくぎょうれつ provides that standard criterion.

The pseudoinverse is a later advanced topic. At this point, the important distinction is that rectangular matrices do not have the same two-sided inverse as square matrices.

6Decision criteria

• Use an inverse matrix逆行列ぎゃくぎょうれつ when you want to recover a map or solve $Ax=b$Ax=b all at once.
• The existence test is whether elimination can turn the left side into $I$I.
• $\det A\ne 0$\det A\ne0, $\mathrm{rank}\left(A\right)=n$\operatorname{rank}(A)=n, and $\ker \left(A\right)=\left\{0\right\}$\ker(A)=\{0\} are the same invertibility condition read from different angles.
• Kernel, rank, unique solutions for every right-hand side, and the existence of an inverse map all express that no information is lost.

7Final forms

Determinants and pseudoinverses are developed later. The direct criteria in this page are elimination, rank, kernel, and unique solvability.

8Theorem and proof: the inverse is unique

Let $B$B and $C$C both be inverses of a square matrix $A$A. That is,

Then $B=C$B=C. The proof is

This uses associativity of matrix multiplication. The proof shows not only that an inverse may or may not exist, but also that if it exists, it is determined uniquely.

9Exercise links

10In one sentence

• An inverse matrix逆行列ぎゃくぎょうれつ exists exactly when the transformation represented by the matrix can be recovered.

11Related links