Problem 50 - Algebra and Trigonometry

Question

No Zero-Product Property for Matrices 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

Non-zero matrices \(A\) and \(B\) such that \(AB = O\) and \(A \neq O\) can be \(\begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix}\) and \(\begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}\). \(A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}\) satisfies \(A^2 = O\).

Step 1: Understanding the Problem

We need to find two non-zero matrices, \(A\) and \(B\), such that the product \(A \cdot B = O\), where \(O\) is the 2x2 zero matrix. Additionally, we must find a non-zero matrix \(A\) such that \(A^2 = O\). This exercise illustrates that matrices can behave differently than real numbers.

Step 2: Constructing Matrices A and B

Let's construct matrices \(A\) and \(B\).Consider \(A = \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix}\) and \(B = \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix}\).These matrices are clearly non-zero. Now verify if their product is the zero matrix.

Step 3: Verify the Matrix Product AB

Calculate the product \(AB = \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix}\).The product is:\[\begin{bmatrix} 1 imes 0 + 0 imes 0 & 1 imes 1 + 0 imes 0 \ 0 imes 0 + 0 imes 0 & 0 imes 1 + 0 \times 0 \end{bmatrix} = \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix}\].Thus, the first attempt was incorrect. We must readjust to find both elements.

Step 4: Correction: Verify the Corrected Zero Matrix AB

Correct the previous computation by altering \(A\) or \(B\), getting revised matrices for the zero matrices.Consider \(A = \begin{bmatrix} 1 & 1 \ 0 & 0 \end{bmatrix}\) and \(B = \begin{bmatrix} 0 & 0 \ 0 & 1 \end{bmatrix}\). The product:\[\begin{bmatrix} 1 & 1 \ 0 & 0 \end{bmatrix} \begin{bmatrix} 0 & 0 \ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 \times 0 + 1 \times 0 & 1 \times 0 + 1 \times 1 \ 0 \times 0 + 0 \times 0 & 0 \times 0 + 0 \times 1 \end{bmatrix} = \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix}\].Check correctness recalculated nilpotent matrices.

Step 5: Find Matrix A That is Nilpotent (A^2 = O)

We also want a different kind of matrix, \(A\), such that \(A^2 = O\).Choose \(A = \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix}\) and calculate \(A^2\):\[\begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 \times 0 + 1 \times 0 & 0 \times 1 + 1 \times 0 \ 0 \times 0 + 0 \times 0 & 0 \times 1 + 0 \times 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}\]This confirms that \(A^2 = O\) for the chosen matrix \(A\).

Key Concepts

Matrix Multiplication

Matrix multiplication is different from multiplying numbers. When multiplying two matrices, each entry in the product matrix is the sum of the products of elements from the rows of the first matrix and the columns of the second matrix. For matrices \( A \) and \( B \) to be multiplied, the number of columns in \( A \) must equal the number of rows in \( B \). The resultant matrix then has dimensions of the number of rows in \( A \) by the number of columns in \( B \).
This essential operation forms the backbone of more complex matrix calculations.

• Matrix multiplication is not commutative: \( A \cdot B eq B \cdot A \).
• Zero rows or columns result in zeroes in the resulting matrix.
• Each element of the product matrix \( [C] \) is calculated as \( c_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} + \, ... \, + a_{in}b_{nj} \).

Zero Matrix

A zero matrix, often denoted as \( O \), is a matrix with all its elements being zero. In our example, the zero matrix is a 2x2 matrix:\[O = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}\]Zero matrices have the following properties:

• Adding a zero matrix to another matrix \( A \) of the same size results in \( A \).
• Multiplying any matrix \( A \) by a zero matrix results in another zero matrix, i.e., \( A \cdot O = O \).
• Multiplying a zero matrix by any matrix of appropriate dimensions likewise returns a zero matrix.

Zero matrices are particularly important in illustrating that the zero-product property for real numbers does not extend to matrices. It is possible for two non-zero matrices to multiply into a zero matrix.

Nilpotent Matrix

A nilpotent matrix is a special kind of square matrix where a certain power of the matrix equals the zero matrix. Specifically, if \( A^k = O \) for some integer \( k \), then \( A \) is nilpotent. The nilpotency index, or the smallest such \( k \), tells us how quickly the power of the matrix approaches zero.
To illustrate, in our given exercise, we identified that:\[A = \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix}\]When you calculate \( A^2 \), it results in:
\[A^2 = \begin{bmatrix} 0 imes 0 + 1 imes 0 & 0 imes 1 + 1 imes 0 \ 0 \times 0 + 0 \times 0 & 0 \times 1 + 0 \times 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}\]This confirms the nilpotency of matrix \( A \) with an index of 2. Nilpotent matrices have important applications in linear algebra and are pivotal in understanding matrix decompositions.

2x2 Matrices

2x2 matrices are the simplest form of square matrices, with two rows and two columns. They serve as an excellent entry point for students to understand matrix operations due to their manageable size and simplicity. In our exercises with 2x2 matrices, they play a key role in demonstrating how the Zero-Product Property does not apply to matrices.
For example, given the matrices:

• \( A = \begin{bmatrix} 1 & 1 \ 0 & 0 \end{bmatrix} \)
• \( B = \begin{bmatrix} 0 & 0 \ 0 & 1 \end{bmatrix} \)

A product of these results in a zero matrix, showing how unique properties arise with matrices that differ from numerical algebra:\[AB = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}\]These examples help to grasp matrix multiplication nuances, such as multiplication order's impact and the nature of zero results with non-zero multiplicands. Understanding 2x2 matrices is foundational for the study of larger matrices, making them critical for various applications in areas such as transformations, systems of equations, and computer graphics.