Problem 50 - Precalculus: Mathematical for Calculus

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

Matrices \( A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \), \( B = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} \) give \( AB = O \). Matrix \( A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \) gives \( A^2 = O \).

Step 1: Understanding the Problem

The task is to find two non-zero matrices, \( A \) and \( B \), such that their product is a zero matrix \( O \). Additionally, we need to identify a single non-zero matrix \( A \) such that its square is the zero matrix, \( A^2 = O \).

Step 2: Finding Two Non-Zero Matrices AB = O

Consider a matrix \( A = \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix} \) and a matrix \( B = \begin{bmatrix} 0 & 0 \ 1 & 0 \end{bmatrix} \). Compute the product:\[ A \cdot B = \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix} \cdot \begin{bmatrix} 0 & 0 \ 1 & 0 \end{bmatrix} = \begin{bmatrix} (1)(0) + (0)(1) & (1)(0) + (0)(0) \ (0)(0) + (0)(1) & (0)(0) + (0)(0) \end{bmatrix} = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix} = O \]. Thus, \( A eq O \) and \( B eq O \) but \( AB = O \).

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

Consider matrix \( A = \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix} \). Compute \( A^2 \): \[ A^2 = \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix} \cdot \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix} = \begin{bmatrix} (0)(0) + (1)(0) & (0)(1) + (1)(0) \ (0)(0) + (0)(0) & (0)(1) + (0)(0) \end{bmatrix} = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix} = O \]. Thus, \( A eq O \), but \( A^2 = O \).

Key Concepts

matrix multiplication

Matrix multiplication is a fundamental operation in linear algebra. It involves multiplying two matrices to produce another matrix. The key point in matrix multiplication is to ensure that the number of columns in the first matrix matches the number of rows in the second matrix. This allows the operation to be defined and produce a resultant matrix.

• For two matrices, say matrix \( A \) with size \( m \times n \), and matrix \( B \) with size \( n \times p \), the result of \( A \, B \) will be a matrix of size \( m \times p \).
• The entry in the \( i \)-th row and \( j \)-th column of the product result is calculated as the sum of the products of the corresponding entries from the \( i \)-th row of \( A \) and the \( j \)-th column of \( B \).

Matrix multiplication is not commutative, which means \( AB eq BA \) in general. This non-commutativity is crucial when dealing with zero or identity elements in matrix algebra.
In the context of the zero-product property, even if \( AB = O \), it does not necessarily imply \( A = O \) or \( B = O \). This distinction from scalar multiplication creates interesting scenarios in matrix algebra.

zero matrix

The zero matrix is a special kind of matrix where all elements are zero. For any given size, say \( m \times n \), a zero matrix is denoted as \( O \) and can be represented as:\[O = \begin{bmatrix}0 & 0 & \cdots & 0 \0 & 0 & \cdots & 0 \\vdots & \vdots & \ddots & \vdots \0 & 0 & \cdots & 0\end{bmatrix}\]
A zero matrix behaves as a kind of "negligent" element in matrix addition and multiplication:

• In addition, any matrix \( A \) added to a zero matrix of the same dimension remains \( A \): \( A + O = A \).
• In multiplication, a zero matrix behaves more like a "destructive" element. For any matrix \( A \), multiplying it by a compatible zero matrix always results in a zero matrix: \( A \cdot O = O \) and \( O \cdot A = O \).

However, it is important to note that the zero-product property fails with matrices. Two non-zero matrices can multiply to give a zero matrix, as seen in the example where matrices \( A \) and \( B \) are non-zero but their product \( AB = O \). This fact highlights a unique aspect of matrix algebra.

non-zero matrices

A non-zero matrix is any matrix that has at least one element not equal to zero. This characteristic distinguishes it from a zero matrix. Non-zero matrices are prevalent in linear algebra, and they appear in various mathematical and applied contexts.
The significance of non-zero matrices in relation to matrix multiplication lies in the failure of the zero-product property for matrices. The zero-product property states that if the product of two numbers is zero, then at least one of the numbers must be zero. However, this property does not hold for matrices.

• For example, given matrices \( A \) and \( B \) where neither is zero, their product \( AB \) can still result in a zero matrix. Such an occurrence challenges intuitive assumptions from scalar mathematics.
• Moreover, a single non-zero matrix can multiply by itself (squared) to yield a zero matrix, such as the matrix \( A \) in the exercise that satisfies \( A^2 = O \).

These peculiarities of non-zero matrices in multiplication make understanding and handling them correctly crucial in advanced linear algebra.