Problem 50 - Algebra and Trigonometry

Question

We have used the Zero-Product Property to solve algebraic equations. Matrices do not have this property. Let O represent the \(2 \times 2\) zero matrix: $$O=\left[\begin{array}{ll}{0} & {0} \\ {0} & {0}\end{array}\right]$$ Find \(2 \times 2\) matrices \(A \neq O\) and \(B \neq O\) such that \(A B=O\) . Can you find a matrix \(A \neq O\) such that \(A^{2}=O ?\)

Step-by-Step Solution

Answer

Matrices \(A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\) and \(B = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}\) satisfy \(AB = O\); Matrix \(A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}\) satisfies \(A^2 = O\).

Step 1: Understanding the Zero Matrix O

The given zero matrix \(O\) is a \(2 \times 2\) matrix with all its elements equal to zero: \(O=\begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}\). We need to find two non-zero matrices \(A\) and \(B\) such that their product is \(O\), and find a non-zero matrix \(A\) such that \(A^2 = O\).

Step 2: Finding Matrices A and B such that AB = O

Let's choose \(A\) and \(B\) such that when multiplied, they result in the zero matrix \(O\). Consider \(A = \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix}\) and \(B = \begin{bmatrix} 0 & 0 \ 0 & 1 \end{bmatrix}\). Calculate the product \(AB = \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix} \begin{bmatrix} 0 & 0 \ 0 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix} = O\), showing that \(AB = O\). Thus, matrices such as these satisfy the first condition.

Step 3: Finding a Matrix A such that A^2 = O

Let's choose \(A = \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix}\). Calculating \(A^2\), we compute \(A \times A = \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix} = O\). Therefore, \(A\) is a matrix where \(A^2 = O\), satisfying the second condition.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental concept in linear algebra. Unlike regular multiplication of numbers, multiplying matrices involves a specific process. For two matrices to be multiplied, the number of columns in the first matrix must be equal to the number of rows in the second matrix. If you have matrices \(A\) and \(B\), where \(A\) is a \(m \times n\) matrix and \(B\) is a \(n \times p\) matrix, the result will produce a \(m \times p\) matrix.

When multiplying matrices, you take each element from the rows of the first matrix and each element from the columns of the second matrix, multiply them, and sum the results. This gives you an entry in the resulting matrix.

• Example: For two \(2 \times 2\) matrices \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\) and \(B = \begin{bmatrix} e & f \ g & h \end{bmatrix}\), the element in the first row, first column of the product \(AB\) is calculated as \(ae + bg\).

Understanding the procedure helps handle problems like finding non-zero matrices \(A\) and \(B\) that multiply to give the zero matrix, as seen in the exercise where different configurations lead to \(AB = O\).

Zero-Product Property

The zero-product property is an important concept in algebra, usually seen with numbers or simple algebraic expressions. It states that if the product of two numbers is zero, at least one of the numbers must be zero. However, this property does not hold for matrices.

In the context of matrices, it is possible for two non-zero matrices to have a product that results in the zero matrix. This is what makes matrix multiplication interesting and slightly different as compared to simple arithmetic multiplication.

• The exercise shows that even when \(A\) and \(B\) are both non-zero \(2 \times 2\) matrices, their product can result in the zero matrix.
• This behavior challenges the intuition developed from scalar multiplication and highlights unique properties of linear transformations that matrices can represent.

Recognizing this important distinction is a key part of understanding linear algebra and how matrices can behave in diverse ways.

2x2 Matrices

Understanding \(2 \times 2\) matrices can provide a useful foundation for matrix algebra. A \(2 \times 2\) matrix has two rows and two columns. This small size makes it ideal for learning basic concepts because calculations stay manageable.

Here's why they are so instructional:

• They allow exploration of matrix properties such as inverse, determinant, and eigenvalues without overwhelming complexity.
• Their computation forms the building blocks for understanding larger matrices, as the basic principles apply to all sizes.
• They are frequently used in solving systems of equations and represent simple linear transformations in two-dimensional space.

For the exercise, \(2 \times 2\) matrices like \(A\) and \(B\) are tested to see how they can be multiplied to form the zero matrix, especially highlighting the fact that certain configurations, although both \(A eq O\) and \(B eq O\), can still yield \(AB = O\). This simple matrix size can reveal complex properties, helping learners grasp fundamental ideas necessary for advancing in matrix theory.