How to find determinants of order 3

Concept 1: Determinants of Order 1 and 2

To every square matrix $A = [a_{ij}]$ of order $n$, we can associate a number (real or complex) called the determinant of the square matrix $A$.

1. Determinant of a matrix of order 1

Let $A = [a_{11}]$ be a matrix of order 1. The determinant of $A$ is defined to be equal to $a_{11}$.
Symbolically: $|A| = det(A) = a_{11}$.

2. Determinant of a matrix of order 2

Let $A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$ be a matrix of order 2. The determinant is calculated as the product of the diagonal elements minus the product of the off-diagonal elements.

Solved Examples (Order 2)

Example 1: Evaluate the determinant $\Delta = \begin{vmatrix} 2 & 4 \\ -1 & 2 \end{vmatrix}$.

Solution: Using the formula $a_{11}a_{22} – a_{21}a_{12}$:

$$\Delta = (2)(2) – (-1)(4) = 4 + 4 = 8.$$

Example 2: Evaluate $\begin{vmatrix} x & x+1 \\ x-1 & x \end{vmatrix}$.

Solution:

$$\Delta = (x)(x) – (x-1)(x+1) = x^2 – (x^2 – 1) = x^2 – x^2 + 1 = 1.$$

Example 3: Find values of $x$ for which $\begin{vmatrix} 3 & x \\ x & 1 \end{vmatrix} = \begin{vmatrix} 3 & 2 \\ 4 & 1 \end{vmatrix}$.

Solution: Expanding both sides:

$$(3)(1) – (x)(x) = (3)(1) – (4)(2)$$ $$3 – x^2 = 3 – 8 \implies -x^2 = -8 \implies x^2 = 8 \implies x = \pm 2\sqrt{2}.$$

Example 4: If $A = \begin{bmatrix} 1 & 2 \\ 4 & 2 \end{bmatrix}$, then show that $|2A| = 4|A|$.

Solution:

First, $|A| = (1)(2) – (4)(2) = 2 – 8 = -6$. Thus, $4|A| = 4(-6) = -24$.
Now, $2A = \begin{bmatrix} 2 & 4 \\ 8 & 4 \end{bmatrix}$.
$|2A| = (2)(4) – (8)(4) = 8 – 32 = -24$.
Therefore, $|2A| = 4|A|$. (Note: For order $n$, $|kA| = k^n|A|$).

Example 5: Evaluate $\begin{vmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{vmatrix}$.

Solution:

$$\Delta = (\cos \theta)(\cos \theta) – (\sin \theta)(-\sin \theta) = \cos^2 \theta + \sin^2 \theta = 1.$$

Concept 2: Determinant of a Square Matrix of Order 3

The determinant of a $3 \times 3$ matrix is determined by expanding it along any row or any column. This process involves “Minors” and “Cofactors” (the sign convention).

1. Expansion Formula (Along Row 1)

Let $A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}$.

2. Sign Convention (Cofactor Signs)

The sign of the term $a_{ij}$ is given by $(-1)^{i+j}$. For a $3 \times 3$ matrix, the signs follow this checkerboard pattern:

Solved Examples (Order 3)

Example 1: Evaluate $\Delta = \begin{vmatrix} 3 & -4 & 5 \\ 1 & 1 & -2 \\ 2 & 3 & 1 \end{vmatrix}$ by expanding along $R_1$.

Solution:

$$\Delta = 3\begin{vmatrix} 1 & -2 \\ 3 & 1 \end{vmatrix} – (-4)\begin{vmatrix} 1 & -2 \\ 2 & 1 \end{vmatrix} + 5\begin{vmatrix} 1 & 1 \\ 2 & 3 \end{vmatrix}$$ $$= 3(1 – (-6)) + 4(1 – (-4)) + 5(3 – 2)$$ $$= 3(7) + 4(5) + 5(1) = 21 + 20 + 5 = 46.$$

Example 2: Evaluate $\Delta = \begin{vmatrix} 0 & \sin \alpha & -\cos \alpha \\ -\sin \alpha & 0 & \sin \beta \\ \cos \alpha & -\sin \beta & 0 \end{vmatrix}$.

Solution: Expanding along $R_1$:

$$\Delta = 0 – \sin \alpha \begin{vmatrix} -\sin \alpha & \sin \beta \\ \cos \alpha & 0 \end{vmatrix} + (-\cos \alpha) \begin{vmatrix} -\sin \alpha & 0 \\ \cos \alpha & -\sin \beta \end{vmatrix}$$ $$= -\sin \alpha (0 – \sin \beta \cos \alpha) – \cos \alpha (\sin \alpha \sin \beta – 0)$$ $$= \sin \alpha \sin \beta \cos \alpha – \cos \alpha \sin \alpha \sin \beta = 0.$$

Example 3: If $A = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4 \end{bmatrix}$, then show that $|3A| = 27|A|$.

Solution:

$|A| = 1(4 – 0) – 0 + 1(0 – 0) = 4$. So $27|A| = 27 \times 4 = 108$.
$3A = \begin{bmatrix} 3 & 0 & 3 \\ 0 & 3 & 6 \\ 0 & 0 & 12 \end{bmatrix}$.
$|3A| = 3(36 – 0) – 0 + 3(0 – 0) = 3 \times 36 = 108$. Verified.

Example 4: Find values of $x$ if $\begin{vmatrix} x & 2 \\ 18 & x \end{vmatrix} = \begin{vmatrix} 6 & 2 \\ 18 & 6 \end{vmatrix}$.

Solution: Expanding:

$x^2 – 36 = 36 – 36 \implies x^2 – 36 = 0 \implies x^2 = 36 \implies x = \pm 6$.

Example 5: Sarrus Rule Visualization

Solution: For a $3 \times 3$ matrix, diagonals can be used for quick calculation: