Problem 28 - Algebra and Trigonometry with Analytic Geometry

Question

\(\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|+\left|\begin{array}{ll}a & b \\ e & f\end{array}\right|=\left|\begin{array}{cc}a & b \\ c+e & d+f\end{array}\right|\)

Step-by-Step Solution

Answer

The equation is true; both sides are equal when expanded.

Step 1: Understanding the Determinant

First, recall how to calculate the determinant of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\). The formula is \(ad - bc\), a basic operation that involves subtraction of the cross-multiplication of its elements.

Step 2: Calculate Individual Determinants

Let's apply this formula to each determinant in the equation: 1. For \(\begin{vmatrix} a & b \ c & d \end{vmatrix}\), the determinant is \(ad - bc\).2. For \(\begin{vmatrix} a & b \ e & f \end{vmatrix}\), the determinant is \(af - be\).3. For \(\begin{vmatrix} a & b \ c+e & d+f \end{vmatrix}\), we apply the determinant formula: \((a(d+f) - b(c+e))\).

Step 3: Simplify the Right Side

Simplify \((a(d+f) - b(c+e))\) to understand its components:- Expand the terms: \(ad + af - bc - be\).This expression can be reorganized to see individual components \(ad - bc + af - be\).

Step 4: Verify the Equality

Compare both sides:- The left side of the equation is \((ad - bc) + (af - be)\).- The expanded right side is \((ad - bc) + (af - be)\).Thus, the left side exactly matches the right side when expanded.

Key Concepts

2x2 matrices

In mathematics, a 2x2 matrix is a simple grid with two rows and two columns. It looks like this:

• a, b on the first row
• c, d on the second row

Written mathematically, it can be represented as \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \). Each element in the matrix corresponds to a position in this structure. Matrices are useful tools for various calculations in mathematics, physics, and computer science.
The 2x2 matrix is often used in basic linear algebra for transformations and solving simple systems of equations. Understanding how to manipulate matrices, such as calculating determinants, is crucial for higher level math and applications.

algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. It's like a language, but for expressing mathematical ideas.

• Variables like x or y are commonly used.
• Instead of numbers, we often solve for these unknowns.

For matrices, algebra is used to perform calculations like finding the determinant. The determinant is a scalar value that, among other things, can tell us if a matrix can be inverted or not. In algebra, knowing how to calculate determinants helps in solving equations that involve matrices.

matrix operations

Matrix operations are various calculations you can perform on matrices. These include addition, subtraction, and multiplication.

• Matrix addition involves adding the corresponding elements from each matrix.
• Matrix subtraction works similarly; you subtract corresponding elements.

For determinants, the primary operation is calculating the determinant itself. In the case of our exercise, you first calculate the determinant of each matrix separately using the formula \( ad - bc \).
This is one of the simplest matrix operations. Once you understand these operations, solving equations involving multiple matrices becomes more manageable.

linear algebra

Linear algebra focuses on vector spaces and linear mappings between these spaces. It involves studying lines, planes, and subspaces, and is fundamental to understanding matrices.

• It provides methods for solving systems of linear equations.
• Matrices are a key component since they can represent linear transformations.

Linear algebra is crucial in more advanced applications such as computer graphics, engineering, and natural sciences. Within this broad field, determinants play a vital role in understanding the properties of a matrix. They help us understand whether a system of equations has a solution, how vectors are transformed by the matrix, and are integral in applications like eigenvalues and eigenvectors.