Question

Find the value of each determinant. \(\left|\begin{array}{rr}6 & 4 \\ -1 & 3\end{array}\right|\)

Short answer

The determinant is 22.

Step-by-step solution

- Understanding the Formula

To find the determinant of a 2x2 matrix, use the formula: \[ \text{det}\begin{pmatrix}a & b \ c & d\begin{pmatrix} = ad - bc \ \text{In this case}, \ a=6, b=4, c=-1, d=3 \]

- Apply the Values

Plug in the values into the determinant formula: \[ \text{det}\begin{pmatrix} 6 & 4 \ -1 & 3\begin{pmatrix} = 6 \times 3 - 4 \times -1 \]

- Perform the Multiplication

Calculate the products: \[ 6 \times 3 = 18 \ -4 \times -1 = -4 \]

- Final Calculation

Subtract the results from Step 3: \[ 18 - (-4) = 18 + 4 = 22 \]

Key concepts

Understanding a 2x2 Matrix

A 2x2 matrix is a simple array of numbers arranged in two rows and two columns. It's one of the basic building blocks in the field of linear algebra and can represent many things, including systems of linear equations, transformations, and more. Each element of the matrix has a specific position: for instance, in the matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\), 'a' is in the first row and first column, 'b' is in the first row and second column, 'c' is in the second row and first column, and 'd' is in the second row and second column. The arrangement might look like simple numbers, but the information they carry can be incredibly powerful and useful in various applications.

Calculating the Determinant of a Matrix

The determinant of a matrix is a special number that can give insights into many properties of the matrix. For a 2x2 matrix, the determinant can be easily calculated using the formula: \(ad - bc\). In the provided exercise example, the matrix is \[ \begin{pmatrix} 6 & 4 \ -1 & 3 \end{pmatrix} \]. Here, 'a' is 6, 'b' is 4, 'c' is -1, and 'd' is 3. Plugging these values into our formula gives us: \ 6 \times 3 - 4 \times (-1) = 18 - (-4) = 18 + 4 = 22. This number, 22, is the determinant of the matrix. The determinant helps us understand several important properties, such as whether the matrix is invertible (only if the determinant is not zero) and the scaling factor of the linear transformation represented by the matrix.

What is Linear Algebra?

Linear algebra is a branch of mathematics that deals with vectors, vector spaces (also called linear spaces), linear transformations (functions that preserve vector addition and scalar multiplication), and systems of linear equations. Concepts from linear algebra are essential in many scientific fields, including physics, computer science, engineering, and economics. One of the central concepts in linear algebra is the matrix. Matrices can represent data or systems of linear equations, and their properties, like determinants, can provide valuable information. By understanding matrices and their determinants, we can solve linear equations, perform transformations, and much more. Linear algebra provides the foundation for advanced studies in various fields and is a crucial tool for modern technologies like machine learning and data analysis. Understanding the basics, such as a 2x2 matrix and its determinant, sets the groundwork for deeper exploration into this fascinating and widely applicable area of mathematics.