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 of the matrix is 6.

Step 1: Identify Matrix Size

Determine the dimensions of the matrix. The given matrix is a 2x2 matrix, which has two rows and two columns.

Step 2: Use the Formula for Determinant of a 2x2 Matrix

For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated as \( ad - bc \). Identify the elements in the matrix: \( a = 2 \), \( b = 0 \), \( c = 0 \), and \( d = 3 \).

Step 3: Calculate the Determinant

Substitute the values into the formula: \( 2 \cdot 3 - 0 \cdot 0 = 6 - 0 = 6 \). The determinant is 6.

Key Concepts

2x2 matrix

A 2x2 matrix is one of the simplest forms of matrices and is often where students begin their journey of understanding this mathematical concept. In a 2x2 matrix, there are precisely two rows and two columns, which means it has a total of four elements. This small size makes it straightforward to visualize and work with compared to larger matrices.

Typically, a 2x2 matrix is represented as:

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

where \(a, b, c,\) and \(d\) are the elements of the matrix.
To better understand it, consider a grid with two vertical and two horizontal lines, creating four sections, each filled with a number. This forms the basic layout of a 2x2 matrix.

calculation of determinants

The determinant of a matrix is a special number that can be calculated from its elements. For a 2x2 matrix, finding the determinant is relatively straightforward thanks to a specific formula. Determinants are useful in various applications, including solving systems of simultaneous equations, and they play an essential role in linear algebra.

For a given 2x2 matrix:

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

the determinant is calculated using the formula:

• \(ad - bc\)

Let's break it down:

• The product of the first diagonal, \(ad\), is calculated first. These are the elements from the top left to the bottom right of the matrix.
• Then, subtract the product of the second diagonal, \(bc\), which runs from the top right to the bottom left.

So, for example, if \(a = 2, b = 0, c = 0,\) and \(d = 3\), then the determinant calculation would be \(2 \times 3 - 0 \times 0 = 6 - 0 = 6\). Thus, the determinant is 6.

matrix dimensions

The dimensions of a matrix tell us the size and shape of the matrix layout. Understanding the dimensions is crucial for performing calculations, such as finding determinants or executing matrix multiplication. Dimension is defined by two numbers: the number of rows and the number of columns.

For example, a matrix with:

• 2 rows and 2 columns is called a 2x2 matrix.
• 3 rows and 4 columns would be a 3x4 matrix.

Reading dimensions involves stating the number of rows first, followed by the number of columns.
Dimensions are essential because certain operations, like addition or multiplication, require matrices of specific sizes. Only matrices with the same dimensions can be added or subtracted from each other. Meanwhile, for multiplication, the number of columns in the first matrix must match the number of rows in the second.
So, knowing dimensions not only helps in solving mathematical problems but also provides insight into the type of operations that can feasibly be conducted with given matrices.