Calculation rules for the determinant行列式ぎょうれつしき

1Introduction

The determinant行列式ぎょうれつしき is defined for square matrices正方行列せいほうぎょうれつ and detects invertibility可逆性かぎゃくせい and oriented volume scaling体積の伸縮率. For computation, the key is to track how elementary row operations行基本変形ぎょうきほんへんけい and elementary column operations列基本変形れつきほんへんけい change the determinant行列式ぎょうれつしき.

2Terms and definition

The determinant行列式ぎょうれつしき is a number $\det A$\det A associated with a square matrix正方行列せいほうぎょうれつ $A$A. For a $2\times 2$2\times2 matrix行列ぎょうれつ,

For the general $n\times n$n\times n definition, the permutation formula chooses one entry from each row and one entry from each column. The calculation rules below are consequences of multilinearity and alternation.

3Strategy

For a $2\times 2$2\times2 matrix, compute the difference of diagonal products. For $3\times 3$3\times3 and larger matrices, choose cofactor expansion, elementary row operations行基本変形ぎょうきほんへんけい, or elementary column operations列基本変形れつきほんへんけい according to which method creates zeros most easily.

4Intuitive explanation

The determinant行列式ぎょうれつしき measures how much a matrix stretches area or volume. If the value is zero, area or volume collapses to zero, information is lost, and an inverse matrix逆行列ぎゃくぎょうれつ does not exist.

5Rigorous explanation

For

we have $\det A=ad-bc$\det A=ad-bc. For $3\times 3$3\times3 matrices, cofactor expansion reduces the computation to $2\times 2$2\times2 determinants.

The effects of row and column operations are as follows.

5.1Why column operations obey the same rules

View the determinant as a function of its columns:

The determinant is linear in each column and becomes zero when two columns are equal. Therefore,

and

The second term is zero because two columns are equal, so column addition does not change the determinant. Swapping two columns reverses orientation, and algebraically follows from alternation.

6Worked examples

Therefore this matrix行列ぎょうれつ is invertible可逆かぎゃく.

Also,

The second row is three times the first row, so the rows are linearly dependent一次従属いちじじゅうぞく and area collapses to zero.

6.1Checking by elementary column operations

Let

If the columns are swapped, then

so

If the first column is multiplied by $\lambda$\lambda, then

and

If $\lambda$\lambda times the first column is added to the second column, then

and

7Common misunderstandings

• The determinant行列式ぎょうれつしき is not defined for arbitrary rectangular matrices; it belongs to square matrices正方行列せいほうぎょうれつ.
• The determinant is not merely a formula; it records invertibility and volume scaling.
• If changes to the determinant during row operations are not recorded, the final value will be wrong.
• Column operations can compute determinants, but they do not directly preserve the solution set of a system of linear equations連立一次方程式れんりついちじほうていしき.

8Scope

Here the focus is on calculation rules. For a general $n\times n$n\times n matrix, the determinant行列式ぎょうれつしき is characterized by multilinearity in rows or columns, alternation交代性こうたいせい, and $$\det I=1$$\det I=1.

9Final forms

10Theorem and proof: why column addition does not change the determinant行列式ぎょうれつしき

The determinant is linear in each column and becomes zero when two columns are equal. These two facts imply that

does not change the determinant.

Proof. Write $D(C_1,\dots ,C_n)=\det [C_1\cdots C_n]$D(C_1,\ldots,C_n)=\det[C_1\ \cdots\ C_n]. Linearity in the $i$i-th column gives

The second term is zero because two columns are equal. Therefore the determinant is unchanged. This proof derives the calculation rule from multilinearity rather than treating it as a memorized trick.

11Exercise links

12Related links