Problem 6 - Algebra and Trigonometry

Question

1–6 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: Identify Matrix Dimensions

To find the dimension of a matrix, first determine the number of rows and the number of columns. The given matrix is:\[\left[\begin{array}{ll}{1} & {0} \ {0} & {1}\end{array}\right]\]Count the number of horizontal lines of numbers for rows and vertical lines for columns.

Step 2: Count the Number of Rows

Count the number of horizontal lines (rows) in the matrix:1. The first row is \([1, 0]\).2. The second row is \([0, 1]\).There are 2 rows in total.

Step 3: Count the Number of Columns

Count the number of vertical lines (columns) in the matrix:1. The first column consists of \(1\) and \(0\).2. The second column consists of \(0\) and \(1\).There are 2 columns in total.

Step 4: State the Matrix Dimensions

The dimensions of a matrix are given by the number of rows followed by the number of columns. Therefore, the dimension of this matrix is 2 rows by 2 columns, expressed as \(2 \times 2\).

Key Concepts

Matrix Rows

Matrix rows refer to the horizontal sets of values within a matrix. Imagine a matrix as a table. Each line of that table running from left to right is a row. In the given example matrix: \[\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \]we have two rows.

• The first row is \[ [1, 0] \]
• The second row is \[ [0, 1] \]

Rows are typically labeled with indices, starting from 1 upward, so you would refer to these as the first and second rows accordingly.
Rows play an essential role in describing the structure of a matrix, as they are half of the dimension calculation, which is explained further in determining matrix dimensions.
Understanding rows helps you know how data is arranged horizontally, which is crucial for performing operations like addition and subtraction of matrices.

Matrix Columns

Matrix columns are the vertical lines within a matrix. Similar to rows, they help define the structure of a matrix, but instead, they arrange and organize data running from top to bottom. In our example matrix: \[\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \]there are two columns.

• The first column is made up of 1 and 0, formatted as \[ \begin{bmatrix} 1 \ 0 \end{bmatrix} \]
• The second column consists of 0 and 1, shown as \[ \begin{bmatrix} 0 \ 1 \end{bmatrix} \]

Columns are numbered starting from the left, which is very useful when performing matrix operations like multiplication. Understanding columns is critical, especially when dealing with larger matrices or performing operations that depend on specific column interactions, such as finding determinants or solving linear equations.
Columns are the other half of a matrix’s dimensions, helping specify its overall shape.

2x2 Matrix

A 2x2 matrix is a specific type of matrix with two rows and two columns. The dimensions of a matrix are noted by expressing the number of rows by the number of columns, such as 2x2, read as "two by two." This particular matrix size is often encountered when dealing with basic transformations or systems of equations in linear algebra.
For the given matrix example: \[\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \]It's classified as a 2x2 matrix because:

• It has two rows: [1,0] and [0,1]
• It contains two columns
• The structure is evident as each element has a clear position in a 2x2 grid.

2x2 matrices are the simplest forms of square matrices, which mean the number of rows equals the number of columns, and they are particularly important when starting to learn about the identity matrix, determinants, inverse matrices, and more.
These concepts form the basis for understanding dimensionality in more complicated matrices used in higher-level math and applications.