Finding the Determinant of a 3x3 Matrix

Finding the Determinant of a 3×3 Matrix — Moosa Academy

A 3×3 determinant is not a new rule to memorise. It breaks down into three ordinary 2×2 determinants, combined with a fixed pattern of signs.

Concept Notation and the idea

The determinant of a matrix is written either as $\det(A)$ or by placing the entries between two vertical bars.

\det(A) = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}

It is turned into a sum of three 2×2 determinants by expanding along the first row. Each entry of that row is multiplied by a smaller determinant of its own.

Theorem The sign pattern

Take the first-row entries in turn. For each one, cover its own row and column; the four numbers left over form a 2×2 determinant.

First term — positive.

Second term — negative, always.

Third term — positive.

The minus on the second term applies no matter what sign the entry itself carries. Omitting it is the single most common mistake here.

Theorem The 2×2 rule

\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc

The main diagonal minus the other diagonal. Every one of the three smaller determinants is evaluated exactly this way.

Example A worked determinant

Find the determinant of the matrix with rows $(1, 2, 3)$ , $(4, 5, 6)$ and $(7, 8, 9)$ .

Cover the row and column of 1:

\begin{vmatrix} 5 & 6 \\ 8 & 9 \end{vmatrix} = 45 - 48 = -3

Cover the row and column of 2:

\begin{vmatrix} 4 & 6 \\ 7 & 9 \end{vmatrix} = 36 - 42 = -6

Cover the row and column of 3:

\begin{vmatrix} 4 & 5 \\ 7 & 8 \end{vmatrix} = 32 - 35 = -3

First term:

+1 \times (-3) = -3

Second term:

-2 \times (-6) = +12

Third term:

+3 \times (-3) = -9

-3 + 12 - 9

⟹ 0

Notice the second term: the entry is $+2$ , yet the term came out positive because the minus of the pattern met the minus of $-6$ .

Note What a zero determinant means

A determinant of 0 means the matrix has no multiplicative inverse.

Any row or column may be used for the expansion — the first row is simply the most convenient.

A row or column containing zeros is often quicker still, since those terms vanish.

Summary

A 3×3 determinant expands into three 2×2 determinants along the first row.
Cover an entry's row and column; what remains is its 2×2 determinant.
The signs run +, −, + — the middle term is always subtracted.
Each 2×2 determinant is ad − bc.
A determinant of 0 means the matrix has no inverse.