Problem 10 - Precalculus Mathematics for Calculus

Question

State the dimension of the matrix. $$\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]$$

Step-by-Step Solution

Answer

The dimension of the matrix is \(2 \times 2\).

Step 1: Understand the Exercise

The exercise asks for the dimension of a given matrix. Matrices are typically described by their size, which is expressed as `rows x columns`.

Step 2: Analyze the Matrix Structure

Examine the matrix provided: \(\left[\begin{array}{ll} 1 & 0 \ 0 & 1 \end{array}\right]\). It is structured with values arranged in rows and columns.

Step 3: Count the Rows and Columns

Count the number of rows in the matrix. There are 2 rows. Count the number of columns in the matrix. There are 2 columns.

Step 4: State the Matrix Dimension

Express the dimension of the matrix based on the count from the previous step. Since there are 2 rows and 2 columns, the dimension of the matrix is \(2 \times 2\).

Key Concepts

Matrix Structure

Matrices are an essential concept in mathematics, useful for solving systems of equations, among other applications. A matrix is essentially a rectangular arrangement of numbers, symbols, or expressions organized in rows and columns.
Think of a matrix as akin to a spreadsheet where each cell is filled with a specific value. The layout of these values is what forms the matrix structure. Each position within this structure holds meaning, corresponding directly to a row and a column index.
Understanding a matrix's structure is crucial because it determines how operations can be performed on it. A solid grasp of this structure forms the foundation for more complex matrix operations, such as addition, subtraction, and multiplication.

Rows and Columns

When examining a matrix, it's essential to identify its rows and columns. Rows are the horizontal lines of entries in a matrix, while columns are the vertical lines. This clear separation into rows and columns is what allows us to analyze and understand the matrix.

• **Rows** are counted from top to bottom. So, starting with the first one on top as Row 1, the next one below that would be Row 2, and so on.
• **Columns** are counted from left to right. The first column on the left is Column 1, the next one is Column 2, and so forth.

The intersection between a row and a column gives a unique position to each element within the matrix. For example, the element at Row 1, Column 2 will always refer to the same position in the matrix, aiding consistent referencing and operations.

Counting Matrix Size

The dimension of a matrix is given by the number of rows followed by the number of columns. This is often expressed in the form `rows x columns`. Understanding how to count and express matrix size is vital as it describes the scope of the matrix.
To find the size of a matrix, follow these steps:

• Count all the rows. For example, our matrix \(\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\) has 2 rows.
• Count all the columns. In our example, there are also 2 columns.
• Combine these counts to express the matrix size. With 2 rows and 2 columns, we term the matrix size as \(2 \times 2\).

Knowing the size of a matrix helps in determining the potential for performing specific mathematical operations with it. For instance, two matrices can be added or subtracted only if they have the same dimensions. Similarly, understanding matrix dimensions is key in matrix multiplication and more advanced applications.