Problem 5 - Algebra and Trigonometry

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: Understanding the Matrix

We have been given a 2x2 matrix: \[ \begin{bmatrix} 2 & 0 \ 0 & 3 \end{bmatrix} \] We need to find the determinant of this matrix.

Step 2: Formula for Determinant of a 2x2 Matrix

For any 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is given by the formula \( a \cdot d - b \cdot c \). In our case, \( a = 2 \), \( b = 0 \), \( c = 0 \), and \( d = 3 \).

Step 3: Calculating the Determinant

Substitute the values into the determinant formula: \[ \text{Determinant} = (2 \cdot 3) - (0 \cdot 0) \] Simplify the expression: \[ \text{Determinant} = 6 - 0 = 6 \] The determinant of the matrix is 6.

Key Concepts

Understanding a 2x2 Matrix

A matrix is essentially a rectangular array of numbers, organized in rows and columns. A 2x2 matrix, as the name suggests, consists of 2 rows and 2 columns. It's one of the simplest types of matrices and serves as a fundamental building block in linear algebra. Let's take a closer look at the matrix given in the problem: \[ \begin{bmatrix} 2 & 0 \ 0 & 3 \end{bmatrix} \]This matrix has the following elements:- The first row consists of the numbers 2 and 0.- The second row consists of 0 and 3.Matrices are often used to represent linear transformations, and the numbers inside can represent various properties depending on their application. Understanding how to work with a simple 2x2 matrix lays the groundwork for handling more complex matrices later on.

Applying the Matrix Determinant Formula

The determinant of a matrix is a special number that can be calculated from its elements. It plays a crucial role in various applications such as understanding matrix properties and solving linear equations. For a 2x2 matrix, the formula for the determinant is straightforward. Consider any 2x2 matrix:\[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \]The determinant is calculated using the formula:\[ \text{Determinant} = a \cdot d - b \cdot c \]This involves multiplying the diagonal elements (from top left to bottom right) and subtracting the product of the other two elements (from top right to bottom left). Applying this formula to our given matrix:- \( a = 2 \), \( b = 0 \), \( c = 0 \), and \( d = 3 \).- Substitute into the formula: \( 2 \cdot 3 - 0 \cdot 0 = 6 - 0 \).Thus, the determinant of the matrix is 6. This simple calculation helps in exploring more complex matrix behaviors.

Interconnections with Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and matrices. It provides the foundations for understanding complex systems and solving equations that involve multiple variables. Determinants, like the one calculated for a 2x2 matrix, are crucial in linear algebra since they provide insight into whether a matrix has an inverse, which is essential for solving systems of equations. A non-zero determinant, like the one we found (6), indicates that the matrix is invertible, that is, there is another matrix which can multiply with it to give the identity matrix. Some key points about the relationship between determinants and linear algebra:

• If the determinant of a matrix is zero, it is singular and does not have an inverse.
• The determinant is used in computing eigenvalues, which help in various applications such as stability analysis and transformations.
• Understanding matrices and determinants is pivotal in calculus, physics, computer graphics, and more.

Mathematics students frequently encounter matrices and determinants as part of linear algebra coursework, making these concepts a staple of their mathematical toolkit.