Problem 5 - Precalculus: Mathematics for Calculus

Question

Find the determinant of the matrix, if it exists. $$\left[\begin{array}{ll} 2 & 0 \\ 0 & 3 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is 6.

Step 1: Identify the Type of Matrix

The given matrix is a 2x2 matrix, \[ \begin{bmatrix} 2 & 0 \ 0 & 3 \end{bmatrix} \] which means it has 2 rows and 2 columns. A determinant can be found for a square matrix, so we can proceed to find it for this matrix.

Step 2: Use the Determinant Formula for 2x2 Matrix

For a 2x2 matrix, the determinant is calculated using the formula: \[\text{det}(A) = ad - bc \] where \[ a, b, c, \text{ and }d \] are the elements of the matrix \[\begin{bmatrix} a & b \ c & d \end{bmatrix}. \] For our matrix, \[ a = 2, b = 0, c = 0, \text{ and } d = 3. \]

Step 3: Substitute the Elements into the Formula

Substitute the given elements into the formula: \[ \text{det}(A) = (2)(3) - (0)(0) \] which simplifies to \[ 6 - 0. \]

Step 4: Simplify the Expression

Calculating the expression from the substitution gives \[ \text{det}(A) = 6. \] Thus, the determinant of the matrix is 6.

Key Concepts

2x2 Matrix

Every matrix of size 2x2 is composed of two rows and two columns. This configuration means there are four elements in total, arranged in a specific grid-like pattern.
A general form of a 2x2 matrix looks like this:

• \(\begin{bmatrix}a & b \c & d\end{bmatrix}\)

Here, each letter represents an individual element of the matrix. The first row contains elements \( a \) and \( b \), while the second row contains \( c \) and \( d \).
2x2 matrices are fundamental in linear algebra and matrix computations.
They are often used to simplify calculations, making it easy to compute determinants and understand matrix operations.

Matrix Algebra

Matrix algebra is a powerful mathematical tool used to perform operations on matrices, such as addition, subtraction, and multiplication.
To perform these operations, certain rules specific to matrices must be followed.

• **Addition/Subtraction**: Requires two matrices of the same dimension. Elements are added or subtracted at corresponding positions.
• **Multiplication**: More complex than addition. Involves multiplying rows by columns, and it is not commutative – meaning \( AB eq BA \) in general.

For a 2x2 matrix, these operations become a foundation for more advanced discussions in linear transformations and systems of linear equations.
Working with matrices allows for efficient computations in fields ranging from computer graphics to engineering.

Calculating Determinants

Calculating the determinant is a crucial concept in matrix algebra. It is a scalar value that can determine several important properties of a matrix, such as invertibility.
For a 2x2 matrix, the formula to compute the determinant is simple:

• \[det(A) = ad - bc\]

Here, \( a, b, c, \) and \( d \) are the elements from the matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \).
The determinant provides insight into the matrix, such as whether it has an inverse (a determinant of zero implies the matrix is non-invertible).
This calculation can be completed in a few steps, ensuring it's both efficient and straightforward, especially for smaller matrices.
Understanding determinants is fundamental in applications involving eigenvalues, solving linear equations, and more.