Problem 27 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

The equation holds true after calculations are verified, confirming its correctness.

Step 1: Understand the Determinant Formulas

For a 2x2 matrix \( \left|\begin{array}{cc} x & y \ z & w \end{array}\right| \), the determinant is calculated as \( xw - yz \). We will use this formula to find the determinants for each matrix in our equation.

Step 2: Calculate the Determinant of First Matrix

The first matrix is \( \left|\begin{array}{cc} a & b \ c & d \end{array}\right| \). Calculate its determinant: \( ad - bc \).

Step 3: Calculate the Determinant of Second Matrix

The second matrix is \( \left|\begin{array}{cc} a & e \ c & f \end{array}\right| \). Calculate its determinant: \( af - ce \).

Step 4: Calculate the Determinant of the Final Matrix in the Equation

The matrix on the right side of the equation is \( \left|\begin{array}{cc} a & b + e \ c & d + f \end{array}\right| \). Calculate its determinant: \( a(d+f) - c(b+e) \).

Step 5: Expand and Simplify Determinants

Expand the determinant from Step 4: \( a(d+f) - c(b+e) = ad + af - cb - ce \).

Step 6: Verify the Equation

Rewrite the original equation using calculated determinants: \( (ad - bc) + (af - ce) = ad + af - cb - ce \). Simplify both sides to check equality: \( ad + af - bc - ce = ad + af - cb - ce \). The equation holds true, satisfying: \( ad + af - bc - ce = ad + af - bc - ce \).

Key Concepts

2x2 matrix

A 2x2 matrix is one of the simplest forms of a matrix. It is a rectangular array consisting of two rows and two columns. In mathematical terms, it is often written as follows:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]Here, \(a, b, c,\) and \(d\) are elements of the matrix. Matrices like this are fundamental in various areas of linear algebra as they offer an accessible way to perform operations such as multiplication and finding determinants.
To calculate the determinant of a 2x2 matrix, use the formula \(ad - bc\). This value provides crucial information about the matrix, such as whether it is invertible or not.

• The determinant can tell us whether the matrix has an inverse.
• A zero determinant implies that the matrix does not have an inverse.
• A non-zero determinant means the matrix is invertible.

Understanding these basics of a 2x2 matrix is the first step in exploring deeper concepts in linear algebra.

matrix properties

Matrix properties are essential characteristics and rules that apply to matrices in linear algebra. Understanding these properties makes working with matrices much more manageable and helps simplify complex mathematical expressions and proofs.
Some important properties of matrices include:

• Commutative Property of Addition: For any two matrices \(A\) and \(B\), the sum \(A + B = B + A\).
• Associative Property of Addition: For any matrices \(A, B,\) and \(C\), \((A + B) + C = A + (B + C)\).
• Distributive Property: Scalar multiplication over addition is distributed, meaning \(k(A + B) = kA + kB\) for any scalar \(k\) and matrices \(A\) and \(B\).
• Determinants Multiply Under Matrix Multiplication: For any two square matrices \(A\) and \(B\) of the same size, \(\det(AB) = \det(A) \cdot \det(B)\).

Matrix properties also include specifics about inverse matrices, eigenvalues, and rank—all of which play a substantial role in advanced linear algebra. By internalizing these properties, you'll gain a deeper understanding of how matrices behave and interact.

linear algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear transformations. It provides the foundation for complex systems and is used in a variety of fields such as computer science, engineering, physics, and data science.
Linear algebra involves concepts such as:

• Vectors: These are quantities defined by their magnitudes and directions.
• Matrix Operations: Includes addition, subtraction, multiplication, and finding inverses or determinants.
• Systems of Linear Equations: These systems can be expressed in matrix form, providing a clear method for finding solutions.
• Eigenvalues and Eigenvectors: Key components in understanding linear transformations and stability.

A strong grasp of linear algebra allows you to approach problems involving multi-dimensional data sets, perform transformations, and solve equations efficiently. It's not just theoretical but incredibly practical, bridging the gap between abstract mathematical concepts and real-world applications.