Problem 29 - Advanced Engineering Mathematics

Question

Determine the size of the matrix \(\mathbf{A}\) such that the given product is defined. $$ \left(\begin{array}{llll} 2 & 1 & 3 & 3 \\ 9 & 6 & 7 & 0 \end{array}\right) \mathbf{A}\left(\begin{array}{l} 0 \\ 5 \\ 7 \\ 9 \\ 2 \end{array}\right) $$

Step-by-Step Solution

Answer

The matrix \( \mathbf{A} \) is \( 4 \times 5 \).

Step 1: Analyze the matrix to the left of \( A \)

Examine the matrix that precedes \( \mathbf{A} \). This matrix has 2 rows and 4 columns. We denote it as a \( 2 \times 4 \) matrix.

Step 2: Analyze the matrix to the right of \( A \)

Examine the matrix that follows \( \mathbf{A} \). This matrix has 5 rows and 1 column. We denote it as a \( 5 \times 1 \) matrix.

Step 3: Determine the conditions for matrix multiplication

For the matrix multiplication \( X \mathbf{A} Y \) to be defined, the matrix \( X \) must have as many columns as \( \mathbf{A} \) has rows, and \( \mathbf{A} \) must have as many columns as \( Y \) has rows.

Step 4: Determine the dimensions of matrix \( A \)

To satisfy the matrix multiplication requirements, \( \mathbf{A} \) must have 4 rows (as it follows \( \mathbf{X} \)) and 5 columns (as it precedes \( Y \)). Therefore, \( \mathbf{A} \) is a \( 4 \times 5 \) matrix.

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial in linear algebra, particularly when dealing with matrix multiplication. A matrix is a rectangular array of numbers arranged in rows and columns. The size or dimensions of a matrix are given in rows and columns form, such as "m by n" where "m" stands for the number of rows and "n" stands for the number of columns.
For example, the matrix given on the left of our exercise is a 2 by 4 matrix because it has 2 rows and 4 columns. Similarly, the matrix on the right is a 5 by 1 matrix as it has 5 rows and a single column. Recognizing and stating these dimensions accurately is essential. It also lays the groundwork for determining if matrix operations can be performed.
When performing matrix multiplication, the inner dimensions must match. This means the number of columns in the first matrix must equal the number of rows in the second matrix. Knowing this allows you to predict the size of the resulting matrix.

Matrix Product

The matrix product is essentially a way to multiply two matrices, given certain conditions are met. This is a fundamental operation in linear algebra. To find a matrix product, each element in the resulting matrix is the sum of products of elements from rows of the first matrix and columns of the second matrix.
In our exercise, for the multiplication \( X \mathbf{A} Y \) to be defined, the dimensions of \( \mathbf{A} \) must allow it to connect the output of \( X \) with the input of \( Y \). Matrix \( X \) needs a partner \( \mathbf{A} \) whose row count matches the column count of \( X \), and whose column count matches the row count of \( Y \).
This defines \( \mathbf{A} \)'s dimensions automatically once \( X \) and \( Y \) are known, making the matrix product not just a calculation task but an essential conformity check of matrix dimensions.

Linear Algebra

Linear algebra is a branch of mathematics that studies vectors, vector spaces, and linear equations. It provides a strong framework for performing operations such as matrix addition, multiplication, and finding determinants and inverses.
Understanding linear algebra concepts is vital to understand why and how matrices can be used to represent and solve systems of linear equations. In the context of matrix multiplication, linear algebra facilitates understanding how transformations represented by matrices are applied sequentially.
This field greatly aids in visualizing data transformations and manipulations, which have practical applications in areas such as computer graphics, machine learning, and engineering.
In our matrix multiplication exercise, we witness linear algebra's power in determining how to appropriately dimension matrices for successful multiplication and understanding the implications of combining these mathematical structures.