Evaluation of 2x2 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 determining the area of parallelograms in vector spaces. The determinant of a 2x2 matrix is the simplest case and is relatively straightforward to compute.

Definition of a 2x2 Determinant

For a 2x2 matrix $A$, which is represented as:

$$ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} $$

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

$$ \det(A) = ad - bc $$

Important Points and Differences

Aspect	Description
Notation	The determinant of a matrix $A$ is denoted by $
Formula	For a 2x2 matrix, the determinant is found by the product of the diagonals minus the product of the off-diagonals.
Geometric Interpretation	The absolute value of the determinant of a 2x2 matrix can be interpreted as the area of the parallelogram formed by the column vectors of the matrix.
Invertibility	A 2x2 matrix is invertible if and only if its determinant is non-zero.
Linear Dependence	If the determinant of a 2x2 matrix is zero, its columns (or rows) are linearly dependent.

Examples

Example 1: Basic Calculation

Let's calculate the determinant of the following matrix:

$$ B = \begin{bmatrix} 3 & 4 \ 2 & 5 \end{bmatrix} $$

Using the formula for the determinant of a 2x2 matrix, we have:

$$ \det(B) = (3)(5) - (4)(2) = 15 - 8 = 7 $$

Example 2: Geometric Interpretation

Consider the matrix:

$$ C = \begin{bmatrix} 1 & 3 \ 2 & 6 \end{bmatrix} $$

The determinant of $C$ is:

$$ \det(C) = (1)(6) - (3)(2) = 6 - 6 = 0 $$

Since the determinant is zero, the area of the parallelogram formed by the column vectors of $C$ is also zero. This indicates that the column vectors are linearly dependent and lie on the same line.

Example 3: Invertibility

For the matrix:

$$ D = \begin{bmatrix} -1 & 2 \ 3 & -6 \end{bmatrix} $$

The determinant is:

$$ \det(D) = (-1)(-6) - (2)(3) = 6 - 6 = 0 $$

Since the determinant is zero, the matrix $D$ is not invertible.

Example 4: Using Determinants to Solve a System of Equations

Consider the system of linear equations:

$$ \begin{align*} ax + by &= e \ cx + dy &= f \end{align*} $$

We can represent this system as a matrix equation $AX = B$, where:

$$ A = \begin{bmatrix} a & b \ c & d \end{bmatrix}, \quad X = \begin{bmatrix} x \ y \end{bmatrix}, \quad B = \begin{bmatrix} e \ f \end{bmatrix} $$

To solve for $X$, we need to find the inverse of $A$, which is possible only if $\det(A) \neq 0$. If $\det(A) = 0$, the system may have no solution or infinitely many solutions.

Conclusion

The evaluation of 2x2 determinants is a fundamental concept in linear algebra. It is a simple yet powerful tool that can provide insights into the properties of a matrix, such as invertibility and linear dependence of its columns or rows. Understanding how to compute and interpret 2x2 determinants is essential for anyone studying linear algebra or related fields.