Problem 13 - Precalculus with Limits

Question

In Exercises 5-20, find the determinant of the matrix. \(\left[ \begin{array}{r} 2 && 6 \\ 0 && 3 \end{array} \right]\)

Step-by-Step Solution

Answer

The determinant of the matrix is 6.

Step 1: Identify the elements of the matrix

For the 2x2 matrix \(\left[ \begin{array}{r} 2 && 6 \ 0 && 3 \end{array} \right]\), there are four elements, namely: a=2, b=6, c=0, d=3.

Step 2: Apply the formula for the determinant of a 2x2 matrix

The formula for the determinant of a 2x2 matrix is ad-bc. If we substitute the values of a, b, c, and d from the matrix, the result is 2*3 - 6*0.

Step 3: Simplify the expression

Solve the given expression, it becomes 6-0 which equals to 6.

Key Concepts

Matrix Algebra

Matrix algebra is a significant branch of mathematics that deals with the study and manipulation of matrices. Matrices are rectangular arrays of numbers, symbols, or expressions, arranged in rows and columns. The individual items in a matrix are called its elements.

In the context of matrix algebra, various operations can be performed including addition, subtraction, multiplication, and finding the determinant. The determinant of a matrix is a special scalar value that can be computed from its elements and provides important properties of the matrix, such as its invertibility. A matrix is invertible if and only if its determinant is not zero.

2x2 Matrix

Determinant of a 2x2 Matrix

The determinant of a 2x2 matrix, which is the simplest non-trivial matrix size, is particularly easy to calculate. The 2x2 matrix is written as:
\[\begin{array}{c c}a & b \c & d \end{array}\]
The determinant of this matrix structure is found using the formula: \(ad - bc\). This formula is a direct application of matrix algebra and is fundamental in precalculus and beyond. It is also used in various applications, including solving systems of linear equations, and in understanding concepts such as eigenvalues and eigenvectors.

Example Exercise

Using the example from the exercise, we have a matrix with elements a=2, b=6, c=0, d=3. Plugging these values into the determinant formula gives us:\(2 \times 3 - 6 \times 0 = 6\). Hence, the determinant is 6. This result can also tell us information about the matrix, like its invertibility. Since the determinant is non-zero, this particular matrix is invertible.

Precalculus

Precalculus is a course that prepares students for calculus, which is usually the gateway to more advanced topics in mathematics. It covers a range of topics including functions, complex numbers, and the various properties and operations of matrices.

Understanding the determinant of a matrix is a critical part of precalculus, as it sets the stage for further study in linear algebra and calculus. It represents an important step in developing mathematical reasoning and problem-solving skills. Furthermore, the logic and techniques learned in computing determinants are foundational for topics such as derivatives, integrals, and multivariable calculus that students will encounter later.