Determinant of a 3 × 3 Matrix

When working with matrices, the determinant is one of the most important concepts you'll encounter. It is a scalar value that can provide valuable insights about the matrix, such as whether it's invertible, and is used in various applications, including solving systems of linear equations, finding eigenvalues, and understanding geometric transformations. In this blog, we’ll explore how to calculate the determinant of a 3 × 3 matrix, its significance, and its real-world applications.

1.0What is a Determinant?

The determinant of a square matrix is a scalar value that is calculated from its elements and provides crucial information about the matrix. For a 3 × 3 matrix, the determinant helps determine whether the matrix has an inverse (i.e., is non-singular) or whether it's singular (i.e., not invertible). Additionally, the determinant plays a key role in linear algebra, geometry, and other fields of mathematics.

2.0Formula for Determinant of a 3 × 3 Matrix

Let’s say we have a 3 × 3 matrix A as follows:

$A=\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|$

To calculate the determinant of this matrix, we use the following formula:

$det(A)=a_{11}.\left|\begin{array}{cc}{a}_{22}& a_{23}\\ a_{32}& a_{33}\end{array}\right|-a_{12}.\left|\begin{array}{cc}{a}_{21}& a_{23}\\ a_{31}& a_{33}\end{array}\right|+a_{13}.\left|\begin{array}{cc}{a}_{21}& a_{22}\\ a_{31}& a_{32}\end{array}\right|$

Here, each of the terms involves a 2x2 matrix (called a minor), and we compute the determinant of these minors. The signs alternate as you move from one term to the next.

3.0Solved Example for Determinant of a 3 × 3 Matrix

Let’s go through a simple example:

$A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$

We’ll calculate the determinant of this matrix step-by-step.

1. Minor of $a_{11}$: $\left|\begin{array}{cc}5& 6\\ 8& 9\end{array}\right|=(5\times 9)-(6\times 8)=45-48=-3$
2. Minor of $a_{12}$: $\left|\begin{array}{cc}4& 6\\ 7& 9\end{array}\right|=(4\times 9)-(6\times 7)=36-42=-6$
3. Minor of $a_{13}$:$\left|\begin{array}{cc}4& 5\\ 7& 8\end{array}\right|=(4\times 8)-(5\times 7)=32-35=-3$

Now, substitute these values into the determinant formula:

$det(A)=1\times (-3)-2\times (-6)+3\times (-3)$

$det(A)=-3+12-9=0$

So, the determinant of matrix A is 0. This means that the matrix is singular (non-invertible).

Example 2: Find the determinant of 3 × 3 matrix

$A=\left[\begin{array}{ccc}2& 3& 1\\ 5& 1& 4\\ 3& 2& 6\end{array}\right]$

Solution:

1. Minor of a₁₁

$\left|\begin{array}{cc}1& 4\\ 2& 6\end{array}\right|=(1\times 6)-(4\times 2)$

= 6 – 8 = –2

2. Minor of a₁₂

$\left|\begin{array}{cc}5& 4\\ 3& 6\end{array}\right|=(5\times 6)-(4\times 3)$

= 30 - 12 = 18

3. Minor of a₁₃

$\left|\begin{array}{cc}5& 1\\ 3& 2\end{array}\right|=(5\times 2)-(3\times 1)$

= 10 –3 = 7

det(A) = 2(–2) –3(18) +1(7) 

= –4 –54 +7

= –51

Example 3: Find the determinant of 3 × 3 matrix

$B=\left[\begin{array}{ccc}3& 5& 2\\ 1& 4& 7\\ 9& 6& 8\end{array}\right]$

Solution: 

1. Minor of a₁₁

$\left|\begin{array}{cc}4& 7\\ 6& 8\end{array}\right|=(4\times 8)-(7\times 6)$

= 32 – 42 = –10

2. Minor of a₁₂

$\left|\begin{array}{cc}1& 7\\ 9& 8\end{array}\right|=(1\times 8)-(7\times 9)$

= 8 - 63 = -55

3. Minor of a₁₃

$\left|\begin{array}{cc}1& 4\\ 9& 6\end{array}\right|=(1\times 6)-(9\times 4)$

= 6 - 36 = -30

det(A) = 3(–10) –5(–55) +2(–30) 

= –30 +275 –60

= 185