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Chapter 3: Problem 63

Evaluate the determinants to verify the equation. $$\left|\begin{array}{ll} w & x \\ y & z \end{array}\right|=-\left|\begin{array}{ll} y & z \\ w & x \end{array}\right|$$

Short Answer

Expert verified
The equation \(\left|\begin{array}{ll} w & x \\ y & z \end{array}\right|=-\left|\begin{array}{ll} y & z \\ w & x \end{array}\right|\) is verified if \(wz - xy = -(yx - wz)\), which simplifies to \(wz - xy = -yx + wz\). When we rearrange this final equation, we get \(0 = 0\), which is always true. Therefore, the equation is verified.

Step by step solution

01

Calculate the determinant of the first matrix

The determinant of the first matrix is given by \(wz - xy\). This is obtained by calculating the product of the elements on the main diagonal \(w\) and \(z\), and then subtracting the product of the other diagonal, which are \(x\) and \(y\).
02

Calculate the determinant of the second matrix

The determinant of the second matrix is computed as \(yx - wz\). Here also, it is computed by multiplying the elements on the main diagonal \(y\) and \(x\), and subtracting the product of the non-diagonal elements \(w\) and \(z\).
03

Verify the equation

To confirm the equation \(\left|\begin{array}{ll} w & x \\ y & z \end{array}\right|=-\left|\begin{array}{ll} y & z \\ w & x \end{array}\right|\), replace the determinants calculated in step 1 and 2. If the determinant of the first matrix equals to the negative of the determinant of the second matrix, then the equation is verified.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Algebra
Linear algebra is a significant branch of mathematics focused on vectors, vector spaces, linear transformations, and systems of linear equations. In essence, it deals with lines and planes, and their relationships, which can be visually represented within a geometric space.

It is foundational for various areas such as physics, engineering, computer science, and even social sciences. The reason for its broad applicability is that it provides a powerful framework for solving equations that describe real-world problems ranging from predicting economic trends to designing complex algorithms in machine learning.
Matrix Determinant
The determinant is a special scalar value that is used in linear algebra to determine the properties of a square matrix. Specifically, in a 2x2 matrix, the determinant is calculated by subtracting the product of the off-diagonal entries from the product of the diagonal entries.

The significance of the determinant extends beyond a mere calculation; it has profound implications. For instance, a nonzero determinant implies an invertible matrix, which further indicates that the system of linear equations has a unique solution. Conversely, a determinant equal to zero suggests a matrix is not invertible and the linear system may have no or an infinite number of solutions.
Properties of Determinants
Determinants possess unique properties that allow mathematicians and scientists to simplify complex calculations. Some of these key properties include:
  • Exchange of rows or columns changes the sign of the determinant.
  • The determinant of a product of matrices equals the product of their determinants.
  • Multiplying one row or column of a matrix by a scalar multiplies the determinant by that scalar.
  • If a matrix has two identical rows or columns, the determinant is zero.

Understanding these properties can provide shortcuts in computations and deeper insights into the geometry of the system described by the matrix, such as volume distortion in linear transformations.
Matrix Algebra
Matrix algebra is the study of matrices and the algebraic operations that can be conducted with them, such as addition, subtraction, and multiplication. Additionally, matrix algebra involves special operations such as finding the transpose, determinant, and the inverse of matrices.

The power of matrix algebra becomes evident in solving multiple simultaneous equations, which would be unwieldy by simple algebraic methods. It also serves as a fundamental component of more advanced topics in linear algebra, such as eigenvalues and eigenvectors, which have applications in areas ranging from quantum mechanics to Google's PageRank algorithm.

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Most popular questions from this chapter

Prove that the determinant of an invertible matrix \(A\) is equal to \(\pm 1\) when all of the entries of \(A\) and \(A^{-1}\) are integers. Getting Started: Denote det( \(A\) ) as \(x\) and \(\operatorname{det}\left(A^{-1}\right)\) as \(y\) Note that \(x\) and \(y\) are real numbers. To prove that \(\operatorname{det}(A)\) is equal to \(\pm 1,\) you must show that both \(x\) and \(y\) are integers such that their product \(x y\) is equal to 1 (i) Use the property for the determinant of a matrix product to show that \(x y=1\) (ii) Use the definition of a determinant and the fact that the entries of \(A\) and \(A^{-1}\) are integers to show that both \(x=\operatorname{det}(A)\) and \(y=\operatorname{det}\left(A^{-1}\right)\) are integers. (iii) Conclude that \(x=\operatorname{det}(A)\) must be either 1 or \(-1\) because these are the only integer solutions to the equation \(x y=1\)

Use either elementary row or column operations, or cofactor expansion, to find the determinant by hand. Then use a software program or a graphing utility to verify your answer. $$\left|\begin{array}{rrr}-1 & 3 & 2 \\\0 & 2 & 0 \\\1 & 1 & -1\end{array}\right|$$

Proof Prove that if an \(n \times n\) matrix \(A\) is not invertible, then \(A[\operatorname{adj}(A)]\) is the zero matrix.

Find two \(2 \times 2\) matrices such that \(|A|+|B|=|A+B|\)

Use the determinant of the coefficient matrix to determine whether the system of linear equations has a unique solution. $$\begin{aligned}x_{1}+x_{2}-x_{3}=& 4 \\\2 x_{1}-x_{2}+x_{3}=& 6 \\\3 x_{1}-2 x_{2}+2 x_{3}=& 0\end{aligned}$$
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