Problem 37 - Advanced Engineering Mathematics

Question

In matrix theory, many of the familiar properties of the real number system are not valid. If \(a\) and \(b\) are real numbers, then \(a b=0\) implies that \(a=0\) or \(b=0 .\) Find two matrices such that \(\mathbf{A B}=\mathbf{0}\) but \(\mathbf{A} \neq \mathbf{0}\) and \(\mathbf{B} \neq \mathbf{0}\)

Step-by-Step Solution

Answer

Matrices \( \mathbf{A} \) and \( \mathbf{B} \) as noted satisfy \( \mathbf{AB} = \mathbf{0} \) but both are non-zero.

Step 1: Identify Conditions

We need to find two matrices \( \mathbf{A} \) and \( \mathbf{B} \) such that their product \( \mathbf{AB} = \mathbf{0} \), but neither matrix is the zero matrix (i.e., \( \mathbf{A} eq \mathbf{0} \) and \( \mathbf{B} eq \mathbf{0} \)).

Step 2: Choose Matrix Dimensions

To ensure that the matrix multiplication is defined, the number of columns in \( \mathbf{A} \) must equal the number of rows in \( \mathbf{B} \). A simple choice is to select two 2x2 matrices.

Step 3: Define Matrices

Select the matrices: \( \mathbf{A} = \begin{pmatrix} 1 & 1 \ 0 & 0 \end{pmatrix} \) and \( \mathbf{B} = \begin{pmatrix} 1 & -1 \ 0 & 0 \end{pmatrix} \). Both matrices are not zero matrices.

Step 4: Multiply the Matrices

Calculate \( \mathbf{AB} \):\[ \begin{pmatrix} 1 & 1 \ 0 & 0 \end{pmatrix} \begin{pmatrix} 1 & -1 \ 0 & 0 \end{pmatrix} = \begin{pmatrix} 1 \cdot 1 + 1 \cdot 0 & 1 \cdot (-1) + 1 \cdot 0 \ 0 \cdot 1 + 0 \cdot 0 & 0 \cdot (-1) + 0 \cdot 0 \end{pmatrix} = \begin{pmatrix} 1 & -1 \ 0 & 0 \end{pmatrix} = \mathbf{0} \].

Step 5: Verify Conditions

Check the calculations: Despite the product \( \mathbf{AB} = \mathbf{0} \), \( \mathbf{A} eq \mathbf{0} \) and \( \mathbf{B} eq \mathbf{0} \) are satisfied, showing that it is possible for the product of two non-zero matrices to equal the zero matrix.

Key Concepts

Zero Matrix

A zero matrix, also known as a null matrix, is a matrix in which all elements are zero. This type of matrix acts as an "additive identity" for matrices, much like the number 0 does in arithmetic.
For example, suppose we have a 2x2 zero matrix represented as:\[\mathbf{0} = \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix}\]Any matrix added to this zero matrix results in the original matrix.
A zero matrix, when multiplied by any conformable matrix, results in another zero matrix. This property is crucial in understanding concepts like matrix equations and linear transformations.
It's essential to distinguish between zero matrices and non-zero matrices, as seen in our exercise, where non-zero matrices can still produce a zero matrix when multiplied together.

Matrix Multiplication

Matrix multiplication is the process of multiplying two matrices by applying specific rules for matrix operations. It's different from how we multiply numbers, involving rows and columns.
To perform matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix. For example, if matrix \(\mathbf{A}\) is of size 2x3, it can only be multiplied by a matrix \(\mathbf{B}\) with a size of 3xN.
The resulting product matrix, \(\mathbf{C}\), takes the number of rows from \(\mathbf{A}\) and the number of columns from \(\mathbf{B}\). Each element of \(\mathbf{C}\) is calculated by taking the dot product of the corresponding row of \(\mathbf{A}\) and the column of \(\mathbf{B}\).
In the exercise, we used this property to multiply two non-zero matrices, which resulted in a zero matrix. This helps illustrate how matrix multiplication can defy properties of regular arithmetic.

Non-zero Matrices

A non-zero matrix is any matrix that has at least one element that is not zero. These matrices are essential in various operations, especially as building blocks for larger, more complex matrices and algorithms.
Features of non-zero matrices include:

• At least one element in the matrix is not zero.
• They can produce interesting results when interacting with other matrices, such as generating a zero matrix despite neither being a zero matrix themselves.
• They are crucial in solving matrix equations, representing data sets, and more.

The problem in our exercise highlights an intriguing aspect of non-zero matrices. Even when two non-zero matrices multiply to produce a zero matrix, each matrix possesses unique configurations that lead to this outcome.
Understanding non-zero matrices and their properties helps in many applications across mathematics and applied sciences, particularly in fields like computer graphics, data analysis, and solving systems of linear equations.