Problem 5 - Advanced Engineering Mathematics

Question

In Problems \(1-6\), state the size of the given matrix. $$ \left(\begin{array}{rrrr} 1 & 5 & -6 & 0 \\ 7 & -10 & 2 & 12 \\ 0 & 9 & 2 & -1 \end{array}\right) $$

Step-by-Step Solution

Answer

The matrix size is 3x4.

Step 1: Understand a Matrix

A matrix is a rectangular array of numbers arranged in rows and columns. For a given matrix, the size is determined by the number of rows and columns.

Step 2: Count the Number of Rows

Examine the matrix and count the number of horizontal lines of numbers. In this case, the matrix has 3 rows as there are three horizontal lines where numbers are arranged: \( \begin{bmatrix} 1 & 5 & -6 & 0 \end{bmatrix} \), \( \begin{bmatrix} 7 & -10 & 2 & 12 \end{bmatrix} \), and \( \begin{bmatrix} 0 & 9 & 2 & -1 \end{bmatrix} \).

Step 3: Count the Number of Columns

Examine the matrix again and count the number of vertical lines of numbers. Each row shows four vertically aligned numbers which means the matrix contains 4 columns: \( \begin{bmatrix} 1 & 7 & 0 \end{bmatrix} \), \( \begin{bmatrix} 5 & -10 & 9 \end{bmatrix} \), \( \begin{bmatrix} -6 & 2 & 2 \end{bmatrix} \), and \( \begin{bmatrix} 0 & 12 & -1 \end{bmatrix} \).

Step 4: State the Size of the Matrix

Now that you've identified the number of rows and columns, state the size of the matrix in the form of rows by columns. The matrix has 3 rows and 4 columns, so its size is 3x4.

Key Concepts

Matrix Definition

A matrix is a collection where numbers are arranged systematically into rows and columns. Think of it like a special grid of numbers that helps in various calculations and solving equations.
Understanding what a matrix is essential, as they play a crucial role in different fields like mathematics, physics, and computer science. In simplest terms, a matrix is a rectangular array, meaning it has a shape determined by its rows and columns.

• A matrix can hold any kind of number – whole numbers, decimals, even fractions.
• Each number within the matrix is called an element.
• The position of each element is identified by its row and column number.

Getting familiar with matrix structures helps solve problems related to systems of equations, transformations, and more.

Matrix Rows

Rows in a matrix are horizontal lines of numbers. Visualize rows as the lines stacking from left to right.
When counting matrix rows, you essentially count these horizontal lines. It is crucial to determine how many such lines exist because they define the first part of a matrix's size.

• A row starts at the leftmost element of a horizontal line and extends to the rightmost element.
• The number of rows tells us how tall a matrix is.

In our example matrix, you observe three distinct rows, each composed of four numbers.

Matrix Columns

Columns in a matrix are the vertical lines running from top to bottom. To find how many columns a matrix has, you count these vertical structures.
Columns help in determining how wide a matrix is, and they make up the second part of a matrix's size.

• Each column is formed by stacking numbers vertically from different rows but keeping a consistent position across.
• The count of columns indicates how many numbers you have lined up vertically.

In the given example, there are four columns, showing that four elements can be aligned side-by-side, vertically.

Rectangular Array

A rectangular array is a concept in mathematics that explains the structure of a matrix. It simply means that the arrangement forms a rectangle.
This characteristic is the core reason why matrices have a defined number of rows and columns. Each row and each column align consistently, forming clear boundaries of a rectangular shape.

• The number of rows and columns doesn't need to be the same; hence, a matrix is not limited to being a square.
• Whether you have more rows, more columns, or an equal number of both, the setup remains a rectangle.

In this matrix exercise, noting the rectangular array helps quickly visualize and calculate the dimensions (or size) of the matrix, characterized as 3x4.