Procedure for computing an inverse matrix逆行列ぎゃくぎょうれつ

1Introduction

The standard computation of $A^{-1}$A^{-1} uses row reduction on the augmented matrix拡大行列かくだいぎょうれつ $[A\mid I]$[A\mid I]. Row operations record reversible changes to the equations, and when the left side becomes $I$I, the right side has become $A^{-1}$A^{-1}.

2Terms and definitions

Let $A$A be a square matrix正方行列せいほうぎょうれつ. A matrix $A^{-1}$A^{-1} satisfying

is called the inverse matrix逆行列ぎゃくぎょうれつ of $A$A.

For computation, form the augmented matrix拡大行列かくだいぎょうれつ

and aim, by elementary row operations行基本変形ぎょうきほんへんけい, for

3Strategy

Check whether $A$A can be transformed into $I$I. If it can, applying the same row operations to the right block $I$I gives $A^{-1}$A^{-1}. If it cannot, then $A$A is not invertible可逆かぎゃく.

4Intuitive explanation

An elementary row operation行基本変形ぎょうきほんへんけい corresponds to multiplying by an invertible matrix可逆行列かぎゃくぎょうれつ on the left. If the combined row operation is $E$E and it changes $A$A into $I$I, then

Therefore $E=A^{-1}$E=A^{-1}. Applying the same operations to the right-side $I$I reveals $E$E, so the right block becomes $A^{-1}$A^{-1}.

5Precise explanation

Applying row operations to

is the same as multiplying on the left by an invertible matrix. If the product of the elementary matrices is $E$E, then

If the left block becomes $I$I, then $EA=I$EA=I. For square matrices, the existence of a left inverse implies invertibility, and $E=A^{-1}$E=A^{-1}. Hence

The key facts are that elementary row operations are invertible left multiplications and that the identity matrix is the neutral element for composition.

6Why the procedure works

Each row operation is left multiplication by an invertible可逆かぎゃく elementary matrix. If the combined product of all row-operation matrices is $E$E, then

When $EA=I$EA=I, we have $E=A^{-1}$E=A^{-1}. Thus the right block is exactly the inverse.

7Worked example

Let

Start with the augmented matrix拡大行列かくだいぎょうれつ

Subtract three times the first row from the second row:

Then subtract two times the second row from the first row:

Therefore

8Failure check

If a row of zeros appears on the left before the identity can be formed, the columns of $A$A are dependent and $A$A is not invertible可逆かぎゃく. In that case the augmented computation must stop with the conclusion that no inverse exists.

9Common misunderstandings

• Row operations must be applied to both blocks of $[A\mid I]$[A\mid I].
• The right block is not the inverse until the left block is exactly $I$I.
• Column operations are not the standard inverse algorithm unless variable changes are also tracked.

10Scope

This procedure computes the inverse matrix of a square matrix正方行列せいほうぎょうれつ. For rank-deficient matrices or rectangular matrices, one uses the pseudoinverse matrix擬似逆行列ぎじぎゃくぎょうれつ or the least squares method最小二乗法さいしょうにじょうほう instead.

11Final forms

12Exercise links

13Related links