Determinants of Matrix 3×3 with Examples

Here you will learn how to find determinants of matrix 3×3 with examples.

Let’s begin –

Determinants of Matrix 3×3

If A = \(\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}\) is a square matrix of order 3,

then \(a_{11}(a_{22}a_{33} – a_{23}a_{a32})\) – \(a_{12}(a_{33}a_{21} – a_{23}a_{31})\) + \(a_{13}(a_{32}a_{21} – a_{22}a_{31})\) = 0 is defined as the determinant of A.

i.e.  | A | = \(a_{11}\begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix}\) – \(a_{12}\begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix}\) + \(a_{13}\begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{vmatrix}\)

Example 1 : find the determinant of A = \(\begin{bmatrix} 3 & -2 & 4 \\ 1 & 2 & 1 \\ 0 & 1 & -1 \end{bmatrix}\).

Solution : | A | = \(\begin{vmatrix} 3 & -2 & 4 \\ 1 & 2 & 1 \\ 0 & 1 & -1 \end{vmatrix}\)

\(\implies\) | A | = \(3\begin{vmatrix} 2 & 1 \\ 1 & -1 \end{vmatrix}\) – \((-2)\begin{vmatrix} 1 & 1 \\ 0 & -1 \end{vmatrix}\) + \(4\begin{vmatrix} 1 & 2 \\ 0 & 1 \end{vmatrix}\)

\(\implies\) | A | = 3(-2 – 1) + 2(-1 – 0) + 4(1 – 0) 

= -9 – 2 + 4 = -7

Example 2 : find the determinant of A = \(\begin{bmatrix} 1 & 2 & 3 \\ -4 & 3 & 6 \\ 2 & -7 & 7 \end{bmatrix}\).

Solution :  | A | = \(\begin{vmatrix} 1 & 2 & 3 \\ -4 & 3 & 6 \\ 2 & -7 & 7 \end{vmatrix}\)

\(\implies\) | A | = \(1\begin{vmatrix} 3 & 6 \\ -7 & 9 \end{vmatrix}\) – \(2\begin{vmatrix} -4 & 6 \\ 2 & 9 \end{vmatrix}\) + \(3\begin{vmatrix} -4 & 3 \\ 2 & -7 \end{vmatrix}\)

\(\implies\) | A | = (27 + 42) – 2(-36 – 12) + 3(28 – 6) = 231

Note :

(1) Only square matrices have their determinants. The matrices which are not square do not have determinants.

(2) The determinant of a square matrix of order 3 can be expanded along any row or column.

(3) If a row or a column of a determinant consist of all zeros, then the value of the determinant is zero.

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