Problem 43 - Intermediate Algebra

Question

Verify each of the following. The variables represent real numbers. (a) \(\left|\begin{array}{ll}a & b \\ a & b\end{array}\right|=0\) (b) \(\left|\begin{array}{ll}a & a \\ b & b\end{array}\right|=0\) (c) \(\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|=-\left|\begin{array}{ll}b & a \\ d & c\end{array}\right|\) (d) \(\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|=-\left|\begin{array}{ll}c & d \\ a & b\end{array}\right|\) (e) \(k\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|=\left|\begin{array}{ll}k a & b \\ k c & d\end{array}\right|\) (f) \(k\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|=\left|\begin{array}{cc}k a & k b \\ c & d\end{array}\right|\)

Step-by-Step Solution

Answer

All parts verify the identities or properties of determinants, confirming them as true.

Step 1: Evaluate Determinant of Part (a)

For the determinant \(\left|\begin{array}{ll}a & b \ a & b\end{array}\right|\), we calculate it using the formula \(ad - bc\). Here, \(a = a, b = b, c = a, d = b\), hence the determinant is \(ab - ab = 0\). Therefore, the determinant is zero.

Step 2: Evaluate Determinant of Part (b)

For the determinant \(\left|\begin{array}{ll}a & a \ b & b\end{array}\right|\), using the formula \(ad - bc\), we have \(a \cdot b - a \cdot b = 0\). The determinant is zero here as well.

Step 3: Verify Identity in Part (c)

Given, \(\left|\begin{array}{ll}a & b \ c & d\end{array}\right|\) should equal \(-\left|\begin{array}{ll}b & a \ d & c\end{array}\right|\). Calculate the former as \(ad - bc\) and the latter as \(-(bc - ad)\). Both simplify to \(ad - bc\), confirming the equality.

Step 4: Verify Identity in Part (d)

Check \(\left|\begin{array}{ll}a & b \ c & d\end{array}\right| = -\left|\begin{array}{ll}c & d \ a & b\end{array}\right|\). For the determinant \(ad - bc\), reverse terms yield \(-(ad - bc)\), showing the properties of determinants.

Step 5: Verify Identity in Part (e)

Evaluate \(k\left|\begin{array}{ll}a & b \ c & d\end{array}\right| = \left|\begin{array}{ll}ka & b \ kc & d\end{array}\right|\). The left evaluates as \(k(ad - bc)\), and applying the determinant formula to the matrix on the right also results in \(k(ad - bc)\), ensuring they are identical.

Step 6: Verify Identity in Part (f)

Here, verify \(k\left|\begin{array}{ll}a & b \ c & d\end{array}\right| = \left|\begin{array}{ll}ka & kb \ c & d\end{array}\right|\). Computing both gives \(k(ad - bc)\). Alternatively, simplification of the determinant using scalar properties affirms the expression is valid.

Key Concepts

Matrix Properties

Matrices are a fundamental concept in linear algebra, serving as essential tools in various mathematical computations. Understanding their properties helps simplify complex algebraic expressions.

• Equal Rows or Columns: When a matrix has two identical rows or columns, its determinant equals zero. This property is observed in parts (a) and (b) of the exercise, where matrices have repetitive rows.


• Symmetry and Antisymmetry: Swapping rows or columns alters the determinant's sign. Parts (c) and (d) use this, showing that flipping elements changes the determinant's sign, reflecting antisymmetry.


• Scalar Multiplication: Multiplying a row or column by a scalar is equivalent to multiplying the determinant by that scalar. This is explored in parts (e) and (f), confirming how determinants adjust with scalar factors.

Recognizing these properties enhances computational efficiency and helps in verifying determinant identities in matrix algebra.

Algebraic Expressions

In mathematics, algebraic expressions are combinations of numbers, variables, and operators (like addition and multiplication). Matrices often use algebraic expressions to form equations that we need to solve or simplify.

When handling matrix determinants in this exercise, algebraic expressions take the form of products and differences. For example, the expression for a 2x2 matrix determinant is calculated as \(ad-bc\). This expression represents the difference between the products of diagonal entries.

• Expansion: For larger matrices, determinants can be expanded using cofactor expansion, which involves breaking down larger matrices into simpler parts.


• Simplification: Simplifying algebraic expressions within matrices is crucial for calculating determinants efficiently. Recognizing patterns, like symmetrical or repetitive items, aids simplification.

Understanding how to manipulate and simplify these expressions allows for easier computation and accurate results when determining matrix characteristics.

Determinant Formulas

Determinant formulas are mathematical expressions used to calculate the determinant of matrices. Knowing these formulas for different types of matrices enables efficient computation.

The simplest determinant formula applies to 2x2 matrices, like those in the exercise: \(|A|=ad-bc\). This formula provides a straightforward way to find the determinant by taking the product of diagonal elements and subtracting the product of off-diagonal elements.

• Generalization to Larger Matrices: For larger matrices like 3x3 or 4x4, cofactor expansion or row reduction techniques can be used. Cofactor expansion involves creating smaller matrices, simplifying computation.


• Special Properties: If a matrix is diagonal or triangular (upper or lower), its determinant is the product of its diagonal elements. This property immensely simplifies calculations.

Grasping these formulas and applying them correctly is crucial for solving matrix-related problems across various mathematical fields.