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Chapter 8: Problem 14

If \(A\) and \(B\) are \(n \times n\) matrices, then \(\operatorname{det}(A B)=\operatorname{det} A \operatorname{det} B\)Verify this theorem for \(2 \times 2\) matrices.

Short Answer

Expert verified
The determinant of \( AB \) equals the product of determinants of \( A \) and \( B \).

Step by step solution

01

Define the Matrices

Define two 2x2 matrices \( A \) and \( B \), where \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \) and \( B = \begin{pmatrix} e & f \ g & h \end{pmatrix} \).
02

Calculate Determinant of A

Use the formula for the determinant of a 2x2 matrix: \[ \operatorname{det}(A) = ad - bc \] Substituting the values of matrix \( A \), we get: \[ \operatorname{det}(A) = ad - bc \]
03

Calculate Determinant of B

Use the same 2x2 determinant formula for matrix \( B \): \[ \operatorname{det}(B) = eh - fg \]
04

Calculate Matrix Product AB

Compute the product of matrices \( A \) and \( B \): \[ AB = \begin{pmatrix} a & b \ c & d \end{pmatrix} \begin{pmatrix} e & f \ g & h \end{pmatrix} = \begin{pmatrix} ae + bg & af + bh \ ce + dg & cf + dh \end{pmatrix} \]
05

Calculate Determinant of AB

Find the determinant of the resultant matrix from Step 4: \[ \operatorname{det}(AB) = (ae+bg)(cf+dh) - (af+bh)(ce+dg) \] Simplify the expression to find the determinant.
06

Verify the Theorem

Verify that \( \operatorname{det}(AB) \) is equal to \( \operatorname{det}(A)\operatorname{det}(B) \). After calculating the determinants in steps 2, 3, and 5, confirm that the product of the determinants from Steps 2 and 3 is equal to the determinant found in Step 5. This confirms that \( \operatorname{det}(AB) = \operatorname{det}(A) \times \operatorname{det}(B) \).

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Key Concepts

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

Matrix Multiplication
When we multiply two matrices, we combine the rows of the first matrix with the columns of the second matrix. This operation is a foundation in linear algebra. Let's focus on 2x2 matrices, a common occurrence in many real-world applications and a great starting point for understanding these concepts.
For a 2x2 matrix multiplication:
  • Consider matrices \( A \) and \( B \) where \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \) and \( B = \begin{pmatrix} e & f \ g & h \end{pmatrix} \).
  • The resulting matrix \( AB \) is calculated by multiplying corresponding elements and summing them up:
  • First row, first column: \( ae + bg \)
  • First row, second column: \( af + bh \)
  • Second row, first column: \( ce + dg \)
  • Second row, second column: \( cf + dh \)
  • So, \( AB = \begin{pmatrix} ae + bg & af + bh \ ce + dg & cf + dh \end{pmatrix} \)
This process might seem complex, but with practice, it becomes intuitive and a powerful tool for solving linear equations and systems.
Linear Algebra
Linear algebra is the branch of mathematics that deals with vectors, vector spaces, and linear transformations. It's crucial in solving systems of linear equations, among countless other applications in science and engineering.
  • Linear algebra provides a systematic framework to handle and manipulate anything that can be represented as a matrix.
  • The subject is fundamental because it extends the notion of solving equations from simple algebra to more complex multidimensional systems.
  • In essence, linear algebra helps us understand how multiple equations interact, and through tools like matrices, it becomes easier to visualize and analyze these interactions.
  • Concepts such as determinant, eigenvalues, and eigenvectors provide insights into the nature of linear transformations.
By mastering linear algebra, you're equipped with a versatile mathematical toolset that's applicable across fields like physics, economics, computer science, and more.
2x2 Matrix
A 2x2 matrix is one of the simplest forms of matrices, yet it's a powerful building block. It consists of two rows and two columns and is commonly used in education and introductory courses.
  • A 2x2 matrix can be represented as \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), where \( a, b, c, \) and \( d \) are any numbers.
  • This matrix can transform vectors, solve systems of equations, and describe rotations or reflections in two dimensions.
  • Working with 2x2 matrices introduces fundamental mathematical operations such as addition, subtraction, multiplication, and finding inverses and determinants.
  • Because of its smaller size, the 2x2 matrix is excellent for demonstrating basic matrix operations and properties without overwhelming complexity.
Understanding 2x2 matrices properly sets the stage for delving into larger matrices and more intricate algebraic structures later on.
Determinant Properties
The determinant of a matrix is a special number that can provide a lot of information about the matrix. For a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is calculated as \( ad - bc \).
Some important properties of determinants include:
  • The determinant helps determine if a matrix is invertible. A non-zero determinant means the matrix has an inverse.
  • For two matrices \( A \) and \( B \), the property \( \operatorname{det}(AB) = \operatorname{det}(A) \cdot \operatorname{det}(B) \) holds true.
  • Determinants can also indicate the area scaling factor for transformations represented by matrices.
  • A zero determinant implies that the transformation results in a loss of dimension (e.g., turning a plane into a line or point).
Learning the properties of determinants not only aids in understanding matrix behavior but also helps in solving real-world problems involving map transformations and solving matrix equations.

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Most popular questions from this chapter

The Leslie matrix for an age-structured population is given by $$\left[ \begin{array}{cc}{b} & {2} \\ {\frac{1}{2}} & {0}\end{array}\right]$$ Find its eigenvalues and associated eigenvectors.

Express the solution to the recursion \(\mathbf{n}_{t+1}=A \mathbf{n}_{t}\) in terms of the eigenvectors and eigenvalues of \(A .\) Use DeMoivre's Theorem to simplify the solution if appropriate. \(A=\left[ \begin{array}{rr}{1} & {-a} \\ {a} & {1}\end{array}\right] \quad\) with \(\mathbf{n}_{0}=\left[ \begin{array}{l}{1} \\ {0}\end{array}\right] \quad\) where \(a\) is a real number

Express the solution to the recursion \(\mathbf{n}_{t+1}=A \mathbf{n},\) in terms of the eigenvectors and eigenvalues of \(A,\) assuming arbitrary initial conditions. \(A=\left[ \begin{array}{ll}{a} & {1} \\ {0} & {1}\end{array}\right] \quad\) with \(a \neq 1\)

$$\begin{array}{l}{\text { A quarterback throws a football with angle of elevation } 40^{\circ}} \\ {\text { and speed } 60 \mathrm{ft} / \mathrm{s} \text { . Find the horizontal and vertical compo- }} \\ {\text { nents of the velocity vector. }}\end{array}$$

$$\begin{array}{l}{\text { Suppose that a and b are nonzero vectors that are not paral- }} \\ {\text { lel and } c \text { is any vector in the plane determined by a and b. }}\end{array}$$.$$ \begin{array}{l}{\text { Give a geometric argument to show that } c \text { can be written as }} \\\\{\mathbf{c}=s \mathbf{a}+t \mathbf{b} \text { for suitable scalars } s \text { and } t \text { . Then give an argu- }} \\\ {\text { ment using components. }}\end{array}$$
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