Problem 11 - College Algebra and Calculus: An Applied Approach

Question

Find the determinant of the matrix. $$ \left[\begin{array}{rr} \frac{2}{3} & 0 \\ -1 & 6 \end{array}\right] $$

Step-by-Step Solution

Answer

The determinant of the matrix is 4.

Step 1: Substituting into the determinant formula

Take the values of a, b, c and d and substitute into the formula. Here it will be \((\frac{2}{3} * 6) - (0 * -1)\).

Step 2: Solving the equation

Now perform the operations. First, do multiplication then subtraction. The result will be \(4 - 0\). Solve this to get 4.

Key Concepts

Matrix Algebra

Matrix algebra is a key area of mathematics that deals with the study and manipulation of matrices. Matrices are arrays of numbers or functions arranged in rows and columns, forming a rectangular grid. They are incredibly useful in representing and solving systems of linear equations, performing geometric transformations, and handling data in various fields of science and engineering.

When engaging with matrix algebra, you encounter operations such as matrix addition, scalar multiplication, and importantly, matrix multiplication. These operations follow specific rules that may differ from ordinary arithmetic. For instance, matrix multiplication is not commutative, meaning that the order in which you multiply matrices matters. Furthermore, the operation of taking a determinant, which we will explore more deeply in the next section, is a part of matrix algebra that connects to a matrix's properties and has significant applications in area calculations, finding the inverse of a matrix, and determining the solvability of a system of linear equations.

Determinant Formula

The determinant of a matrix is a special scalar value that serves as an indicator of several properties of the matrix. It can tell you if the matrix is invertible, provide information about the volume scaling factor for geometric transformations, and more. For a 2x2 matrix, which is the case in the exercise provided, the determinant can be easily calculated using a simple formula.

The general determinant formula for a 2x2 matrix is given as \[ \text{det}\left(\left[\begin{array}{cc} a & b \ c & d \end{array}\right]\right) = ad - bc \.\]In the exercise, you substituted values into the determinant formula and followed the specific steps to calculate the determinant of the given matrix. Understanding the formula and following the substitution and simplification steps correctly are crucial to finding the determinant. The ability to calculate determinants is integral to numerous matrix operations and higher-level mathematical concepts like eigenvalues and eigenvectors.

Matrix Operations

Matrix operations encompass various computations involving matrices. These include the aforementioned matrix addition and multiplication, but also extend to finding the transpose, adjugate, and inverse of a matrix. Determinants are part of these operations, as they are essential in computing the inverse of square matrices — without a non-zero determinant, an inverse does not exist.

Determinants are also applied in solving systems of equations using Cramer's Rule, which relies exclusively on determinant calculations to find the solution for each variable. Understanding distinct matrix operations and their properties allows for effective problem-solving within linear algebra and provides the foundation for more advanced applications in numerous mathematical and practical contexts, including three-dimensional modeling, optimization problems, and numerical analysis.