Problem 23 - Precalculus: Functions and Graphs

Question

Verify the Identity by expanding each determinant. $$\left|\begin{array}{ll} a & k b \\ c & k d \end{array}\right|=k\left|\begin{array}{ll} a & b \\ c & d \end{array}\right|$$

Step-by-Step Solution

Answer

The identity is verified since both sides simplify to \( k(ad - bc) \).

Step 1: Compute the Determinants Separately

First, we need to compute each determinant individually. Start with the left side of the equation:\[ \left| \begin{array}{ll} a & kb \ c & kd \end{array} \right| \]The determinant is computed as \( a(kd) - (kb)c \), which simplifies to \( akd - kbc \).

Step 2: Factor out k from the Left-side Determinant

Notice that both terms in the calculation of the determinant \( akd - kbc \) have a common factor of \( k \), so we can factor \( k \) out:\[ k(ad - bc) \]

Step 3: Compute the Right-side Determinant

Now calculate the right side:\[ k \left| \begin{array}{ll} a & b \ c & d \end{array} \right| \]The determination involves:\[ k (ad - bc) \] since \( \left| \begin{array}{ll} a & b \ c & d \end{array} \right| = ad - bc \).

Step 4: Verify the Equality

Now that we have both sides simplified:- Left side: \( k(ad - bc) \)- Right side: \( k(ad - bc) \)Since both sides are identical, the identity \[ \left| \begin{array}{ll} a & kb \ c & kd \end{array} \right| = k \left| \begin{array}{ll} a & b \ c & d \end{array} \right| \] is verified.

Key Concepts

Matrix Algebra

Matrix algebra is a fascinating field of mathematics that involves the manipulation of matrices to understand and solve problems. A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. In algebra, matrices can be added, subtracted, multiplied, and even used to solve systems of linear equations.

The basic operations in matrix algebra include addition and multiplication. However, unlike regular arithmetic, matrix multiplication is not commutative, which means that the order in which you multiply matters.

• Matrix addition: Two matrices of the same dimensions can be added by adding their corresponding elements.
• Matrix multiplication: This involves taking the dot product of rows and columns and is central to many operations involving matrices.

Matrices are widely used in different areas such as physics, computer science, and statistics, due to their capability to handle multiple equations simultaneously through a systematic approach.

Determinant Properties

Determinants are powerful tools in matrix algebra, used primarily to solve systems of linear equations, find the inverse of matrices, and understand the properties of linear transformations. A determinant is a special number that can be calculated from a square matrix.

The main properties of determinants are crucial for simplifying complex matrix operations. They help us understand how matrices behave under certain transformations and manipulations. Here are a few important properties:

• If a matrix has two identical rows or columns, its determinant is zero.
• Switching two rows or columns of a matrix reverses the sign of its determinant.
• The determinant of a triangular matrix (upper or lower) is the product of its diagonal elements.
• Multiplying a row or a column by a constant multiplies the determinant by the same constant.

These properties are instrumental in verifying determinant identities, as seen in your exercise. By understanding and applying these properties, you can simplify determinants and factor them correctly, as shown in the solution provided.

Factoring Determinants

Factoring determinants is an essential skill when working with matrices, especially for simplifying and solving determinants. It refers to the process of breaking down a determinant into simpler components. This can help verify mathematical identities more easily, as demonstrated in the given exercise.

In the exercise example, we noticed that the elements in the determinant could be factored. Since each term had the common factor 'k', we factored 'k' out of the determinant calculation. This is a crucial step because factoring can reduce complex equations into simpler forms that are easier to solve.

• Factor out common elements to simplify calculations.
• Recognize patterns and simplifications through detachment of variables or constants from the determinant expression.

Once a determinant is factored, comparative analysis with other determinants becomes evident and straightforward, as both sides of an equation can be analyzed more effectively. Understanding the rationale behind factoring helps not just in solving particular exercises but also in recognizing efficiencies in matrix calculations overall.