Problem 96 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the determinant(s) to verify the equation. \(\left|\begin{array}{rr}w & x \\ c w & c x\end{array}\right|=0\)

Step-by-Step Solution

Answer

The determinant of the given matrix is indeed 0 which verifies the given equation.

Step 1: Identify the Elements of the Matrix

The matrix is given by \(\left|\begin{array}{rr}w & x \ c w & c x\end{array}\right|\). So, you have \(a=w\), \(b=x\), \(c=cw\), and \(d=cx\)

Step 2: Calculate the Determinant

The determinant of a 2x2 matrix is calculated as \(a \cdot d - b \cdot c\). Substituting the variables, you get: \(w \cdot c x - x \cdot c w\).

Step 3: Simplification

Multiplying these out, both terms become \(wxc\). Taken together you get: \(wxc - wxc = 0\).

Key Concepts

2x2 Matrix

A 2x2 matrix is a square array of numbers with two rows and two columns. It's one of the simplest square matrices, making it easier to grasp basic concepts such as matrix operations. In the context of an example matrix:

• First row: elements are given as \( w \) and \( x \).
• Second row: elements include \( cw \) and \( cx \).

These elements are strategically placed, defining the structure of the matrix. Understanding this layout helps when performing operations like calculating the determinant or performing row actions.
It's essential to be meticulous about the position of each element, as even a minor mistake can lead to incorrect results. The arrangement and properties of these elements are foundational to understanding operations such as matrix multiplication and finding the determinant.

Determinant Calculation

The determinant is a crucial aspect of matrices, especially when dealing with 2x2 matrices. This value helps determine properties such as whether or not a matrix is invertible.In general, the determinant of a 2x2 matrix:\[\left|\begin{array}{cc}a & b \c & d\\end{array}\right|\]is calculated as:\[a \cdot d - b \cdot c\].
For the given matrix, knowing this operation helps in verifying equations like the one presented, where you need to compute:

• \( a = w \)
• \( d = cx \)
• \( b = x \)
• \( c = cw \)

Plugging these into the determinant formula results in \( w \cdot cx - x \cdot cw \).
This measure determines matrix properties, ensuring that each step of simplification progresses accurately.

Matrix Simplification

Simplification in matrices often involves methods that ease calculation, maintaining accuracy while reducing complexity. In the case of our 2x2 matrix, the calculated determinant is \( wxc - wxc \),which simplifies to zero. This happens because:

• Each element \( wxc \) starts as a unique product of the matrix elements.
• Both terms are identical, cancelling each other out.

The result of zero implies specific characteristics, like dependency between rows, where one row might be a multiple of the other. This highlights the significance of simplification.
Being adept at such simplifications provides a solid foundation for more advanced matrix operations, showcasing how elemental arithmetic and algebra intertwine in linear algebra.