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The Foundation of Determinant

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A 2 × 2 determinant is expressed as follows.

$$\left[\begin{array}{cc}a_{11}&a_{12}\\  a_{21}&a_{22}\\  \end{array}\right]=a_{11}a_{22}-a_{21}a_{12}$$

Then a 3 × 3 determinant is expressed as follows.

$$\left[\begin{array}{ccc}a_{11}&a_{12}&a_{13}\\  a_{21}&a_{22}&a_{23}\\  a_{31}&a_{32}&a_{33}\\ \end{array}\right]\\=a_{11}a_{22}a_{33}+a_{21}a_{32}a_{13}+a_{31}a_{23}a_{12}\\-a_{31}a_{22}a_{13}-a_{21}a_{12}a_{33}-a_{11}a_{23}a_{32}$$

So far you can memorize easily, but it is hard to memorize from 4 × 4. In this article, I’ll show you how to find a 4×4 determinant by looking at the elements of the first row.

How to find a 4×4 determinant

$$\left[\begin{array}{cccc}a_{11}&a_{12}&a_{13}&a_{14}\\  a_{21}&a_{22}&a_{23}&a_{24}\\  a_{31}&a_{32}&a_{33}&a_{34}\\  a_{41}&a_{42}&a_{43}&a_{44}\\ \end{array}\right]\\=a_{11}\left[\begin{array}{ccc}a_{22}&a_{23}&a_{24}\\a_{32}&a_{33}&a_{34}\\a_{42}&a_{43}&a_{44}\\\end{array}\right]-a_{12}\left[\begin{array}{ccc}a_{21}&a_{23}&a_{24}\\a_{31}&a_{33}&a_{34}\\a_{41}&a_{43}&a_{44}\\\end{array}\right]\\+a_{13}\left[\begin{array}{ccc}a_{21}&a_{22}&a_{24}\\a_{31}&a_{32}&a_{34}\\a_{41}&a_{42}&a_{44}\\\end{array}\right]-a_{14}\left[\begin{array}{ccc}a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\\a_{41}&a_{42}&a_{43}\\\end{array}\right]$$

With respect to the signs of the 3×3 determinants on the right hand, they are positive if the sum of the numbers of row and column of \(a\) is even, and they are negative if the sum is odd. For example, the coefficient of  the first term on the right side is \(a_{11}\). Since the sum of the numbers of row and column is 1 + 1 = 2 and is an even number. Therefore, the sign of the first term is positive. On the other hand, the coefficient of the second term is \(a_{12}\). Since the sum of the numbers of row and column is 1 + 2 = 3 and is an odd number. Therefore, the sign of the second term is negative.

This method is also useful for finding a determinant with 5 × 5 determinant.

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