Evaluation of 3x3 Determinants

Determinants are mathematical objects that are very useful in linear algebra, particularly in solving systems of linear equations, finding the inverse of a matrix, and describing geometric properties. When it comes to 3x3 matrices, evaluating the determinant can be done through several methods. Below, we will explore the standard method for evaluating 3x3 determinants, as well as some important properties and examples.

Standard Method for Evaluating 3x3 Determinants

Given a 3x3 matrix $A$:

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

The determinant of matrix $A$, denoted as $|A|$ or $\det(A)$, can be calculated using the following formula:

$$ \det(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) $$

This method is known as the rule of Sarrus or the diagonal method. It involves multiplying the elements along the diagonals and subtracting the products of the diagonals that run from bottom-left to top-right from the products of the diagonals that run from top-left to bottom-right.

Example of Evaluating a 3x3 Determinant

Let's evaluate the determinant of the following matrix:

$$ B = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{bmatrix} $$

Using the formula:

$$ \begin{align*} \det(B) &= 1(5 \cdot 9 - 6 \cdot 8) - 2(4 \cdot 9 - 6 \cdot 7) + 3(4 \cdot 8 - 5 \cdot 7) \ &= 1(45 - 48) - 2(36 - 42) + 3(32 - 35) \ &= 1(-3) - 2(-6) + 3(-3) \ &= -3 + 12 - 9 \ &= 0 \end{align*} $$

So, the determinant of matrix $B$ is 0.

Properties of Determinants

When evaluating 3x3 determinants, it's important to be aware of some key properties that can simplify the process:

Property	Description
Linearity	The determinant is a linear function with respect to each row or column.
Permutation	Swapping two rows or columns changes the sign of the determinant.
Multiplication	The determinant of a product of matrices equals the product of their determinants.
Zero Row/Column	If a matrix has a row or column of zeros, its determinant is zero.
Proportional Rows/Columns	If two rows or columns are proportional, the determinant is zero.
Triangular Matrix	The determinant of a triangular matrix (upper or lower) is the product of its diagonal elements.

Additional Methods for Evaluating 3x3 Determinants

Cofactor Expansion

Cofactor expansion, also known as Laplace's expansion, is another method to evaluate determinants. For a 3x3 matrix, it involves picking a row or column and expanding the determinant along that row or column using the minors and cofactors.

Example of Cofactor Expansion

Using the same matrix $B$ from above, let's expand along the first row:

$$ \det(B) = 1 \cdot C_{11} - 2 \cdot C_{12} + 3 \cdot C_{13} $$

Where $C_{ij}$ is the cofactor of element $a_{ij}$, which is calculated as $(-1)^{i+j}M_{ij}$, with $M_{ij}$ being the minor of $a_{ij}$ (the determinant of the 2x2 matrix that remains after removing the $i$-th row and $j$-th column).

Matrix of Minors, Cofactors, and Adjugate

For more complex matrices or when finding the inverse of a matrix, it's often useful to compute the matrix of minors and then the matrix of cofactors. The adjugate (or adjoint) of a matrix is the transpose of the matrix of cofactors, and it plays a role in finding the inverse of a matrix.

Conclusion

Evaluating 3x3 determinants is a fundamental skill in linear algebra. Understanding the standard method, properties of determinants, and alternative methods like cofactor expansion can greatly aid in solving problems involving matrices. Practice with different matrices and methods will help solidify these concepts for exams and further mathematical studies.