Problem 30 - Advanced Engineering Mathematics

Question

Determine the size of the matrix \(\mathbf{A}\) such that the given product is defined. $$ \left(\begin{array}{rrr} 2 & 1 & 3 \\ 3 & 9 & 6 \\ 7 & 0 & -1 \end{array}\right) \mathbf{A}\left(\begin{array}{ll} 0 & 1 \\ 7 & 4 \end{array}\right) $$

Step-by-Step Solution

Answer

Matrix \(\mathbf{A}\) has dimensions \(3 \times 2\).

Step 1: Identify the Dimensions of the First Matrix

The first matrix given is \(\begin{bmatrix} 2 & 1 & 3 \ 3 & 9 & 6 \ 7 & 0 & -1 \end{bmatrix}\). It has 3 rows and 3 columns. Therefore, its dimension is \(3 \times 3\).

Step 2: Identify the Dimensions of the Third Matrix

The third matrix given is \(\begin{bmatrix} 0 & 1 \ 7 & 4 \end{bmatrix}\). This matrix has 2 rows and 2 columns, so its dimension is \(2 \times 2\).

Step 3: Determine the Structure of the Product

For the matrix products to be defined, the number of columns in \(\mathbf{A}\) must match the number of columns in the first matrix, and the number of rows in the third matrix must match the number of rows in \(\mathbf{A}\).

Step 4: Set Up the Requirement for \(\mathbf{A}\)

The first matrix \((3 \times 3)\) is multiplied by \(\mathbf{A}\), which means \(\mathbf{A}\) must have 3 rows. After \(\mathbf{A}\), the matrix product is completed with \((2 \times 2)\) matrix, implying \(\mathbf{A}\) must have 2 columns. Hence, \(\mathbf{A}\) has dimensions \(3 \times 2\) to satisfy both conditions.

Key Concepts

Matrix Dimensions

Understanding the dimensions of matrices is fundamental when dealing with matrix multiplication. The dimensions of a matrix are given by the number of rows and columns it contains, usually expressed as \(m \times n\), where \(m\) is the number of rows and \(n\) is the number of columns.

In the original exercise, we are given three matrices, each with specified dimensions. It's important to note these dimensions because they determine whether or not a matrix multiplication can happen.

• The first matrix is a \(3 \times 3\) matrix, which means it has 3 rows and 3 columns.
• The last matrix is a \(2 \times 2\) matrix, having 2 rows and 2 columns.
• The matrix \(\mathbf{A}\), which we need to determine, plays a crucial role in this multiplication sequence and is central to setting a correct matrix multiplication.

When considering these dimensions, \(\mathbf{A}\) must have 3 rows to allow multiplication with the first matrix. Moreover, \(\mathbf{A}\) must have 2 columns for the product with the third matrix to be defined. Thus, matrix \(\mathbf{A}\) must be \(3 \times 2\). This ensures that the combined multiplication is mathematically feasible.

Matrix Algebra

Matrix algebra deals with the rules and operations involving matrices, including addition, subtraction, and importantly, multiplication. For multiplication, there are specific compatibility rules concerning the dimensions of the matrices involved.

A fundamental criterion for multiplying two matrices is that the number of columns in the first matrix must match the number of rows in the second. If the first matrix is of dimensions \(a \times b\), and the second is \(b \times c\), they can be multiplied to yield a new matrix of dimensions \(a \times c\).

• In the exercise, the first matrix is \(3 \times 3\), and the third is \(2 \times 2\). \(\mathbf{A}\) needs to bridge these dimensions correctly.
• This bridging is why \(\mathbf{A}\) has to be \(3 \times 2\). It must match the column count of the first matrix (3 columns) and the row count of the third matrix (2 rows).

Matrix algebra ensures that operations are executed consistently and predictably, forming the backbone of techniques used in various applications such as solving systems of linear equations.

Matrices

Matrices are a core concept in linear algebra and are used to organize and manipulate data in rows and columns. They can represent systems of linear equations, transformations in space, and even data in computer graphics.

Understanding matrices involves being comfortable with terms such as entries (individual values or numbers in a matrix), rows and columns, and recognizing the structured organization these terms represent.

• The number of entries in a matrix is simply the product of its dimensions. For example, a \(3 \times 3\) matrix contains 9 entries.
• Matrices are not only important in theory but also in practical applications like calculating movements in physics or even in machine learning algorithms.

Being familiar with these concepts and being able to traverse from one matrix operation to another smoothly is essential as you delve deeper into the powerful realm of matrix operations. This exercise serves as a step towards building that foundational understanding.