Problem 48 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate the determinants to verify the equation. $$\left|\begin{array}{cc}w & x \\\c w & c x\end{array}\right|=0$$

Step-by-Step Solution

Answer

The determinant of the given 2x2 matrix is 0

Step 1: Identify the matrix

The given equation involves calculation of the determinant of a 2x2 matrix. The matrix is \[\begin{array}{cc}w & x \c w & c x\end{array}\]

Step 2: Formulate the determinant

For a 2x2 matrix A with elements as \[\begin{array}{cc}a & b \c & d\end{array}\], the determinant is calculated as \(det(A) = ad - bc\). Here, in the given matrix, \(a = w\), \(b = x\), \(c = cw\) and \(d= cx\)

Step 3: Calculate the determinant

Substituting the values in the determinant formula, we get \(det(A) = w*cx - x*cw\). This simplifies to \(det(A) = 0\).

Key Concepts

Determinant Calculation

Understanding how to compute the determinant of a matrix is crucial for various applications in mathematics, including solving systems of linear equations and finding the inverse of a matrix. The determinant of a matrix can be seen as a scalar value that encapsulates unique properties of the matrix.

For a 2x2 matrix, the calculation is straightforward. If we have a matrix \[ A = \begin{array}{cc} a & b \ c & d \end{array} \], the determinant, denoted as \( \det(A) \) or \( |A| \) is calculated by multiplying the elements on the diagonal from the top left to the bottom right, subtracting the product of the elements on the diagonal from the top right to the bottom left. Mathematically, this is expressed as \( \det(A) = ad - bc \).

This operation gives us a single number that, in the case of our exercise, represents certain characteristics related to the linear transformation described by the matrix. When the determinant is zero, it tells us that the matrix does not have an inverse and the transformation it represents is not one-to-one.

2x2 Matrix

A 2x2 matrix consists of four elements arranged in two rows and two columns. This arrangement forms a square grid where each element in the matrix can be identified by its position in the row and column.

The general form of a 2x2 matrix is \[ \begin{array}{cc} a & b \ c & d \end{array} \] where \( a, b, c, \) and \( d \) are elements that can be numbers, symbols, or expressions. The first element \( a \) is located in the first row and first column, often referred to as the '11' position. Similarly, \( b \) is in the '12' position, \( c \) in the '21', and \( d \) in the '22' position.

When working with 2x2 matrices, it's important to remember the significance of each position, as the determinant relies heavily on the placement of these elements. The ease of computation and the simplicity of its form make the 2x2 matrix a fundamental concept in matrix algebra, often serving as an introduction to more complex matrix operations.

Matrix Algebra

Matrix Algebra is a branch of mathematics that deals with matrices and the operations that can be performed on them. It includes operations like addition, subtraction, and multiplication of matrices, as well as more advanced concepts like finding determinants, inverses, and solving systems of linear equations.

In matrix algebra, we often deal with square matrices, such as the 2x2 matrix from our example, because they have special properties like determinants and eigenvalues. However, matrices are not limited to square shapes; they can be rectangular as well. Even so, certain operations like calculating a determinant are exclusive to square matrices.

Understanding matrix algebra is essential for advanced studies in fields such as engineering, physics, and computer science. The ability to manipulate and interpret matrices allows for solving complex problems efficiently, especially when dealing with linear transformations and vector spaces. It's important to approach matrix algebra methodically, ensuring that each step in an operation follows the defined rules and keeping track of the arrangement of elements within the matrix.