Problem 2 - Algebra and Trigonometry

Question

1–6 State the dimension of the matrix. $$\left[\begin{array}{rrrr}{-1} & {5} & {4} & {0} \\ {0} & {2} & {11} & {3}\end{array}\right]$$

Step-by-Step Solution

Answer

The matrix is 2 x 4.

Step 1: Identify Rows

A matrix is composed of rows and columns. The given matrix is:\[\begin{bmatrix}-1 & 5 & 4 & 0 \ 0 & 2 & 11 & 3\end{bmatrix}\] Start by identifying how many rows are present. Each row is a horizontal line of elements.

Step 2: Count Rows

Count the horizontal lines of elements in the matrix. There are two horizontal rows.

Step 3: Identify Columns

Next, identify the columns. Columns run vertically in the matrix. The given matrix appears as: \[\begin{bmatrix}-1 & 5 & 4 & 0 \ 0 & 2 & 11 & 3\end{bmatrix}\]

Step 4: Count Columns

Count the sequence of numbers in each vertical set to determine the number of columns. In this matrix, there are four columns.

Step 5: State Dimension

The dimension of a matrix is given as 'rows x columns'. In this case, the matrix has 2 rows and 4 columns, so its dimension is 2 x 4.

Key Concepts

Understanding Matrix Rows

Matrix rows are horizontal lines where elements are organized from left to right. In any matrix, each row is composed of one or more elements. For example, in the given matrix:\[ \begin{bmatrix} -1 & 5 & 4 & 0 \ 0 & 2 & 11 & 3 \end{bmatrix} \]Each level of horizontal numbers forms a row. Understanding how to recognize these rows is key to determining matrix dimensions.

• The first row in this matrix contains the elements: -1, 5, 4, 0.
• The second row contains the elements: 0, 2, 11, 3.

Counting these horizontal sequences will give you the number of rows in the matrix, which in this case is 2. Rows will help you figure out the first part of a matrix's dimension (rows x columns), representing how many sequences of numbers run horizontally.

Analyzing Matrix Columns

Matrix columns are the vertical stacks of numbers, extending from top to bottom. Identifying them is crucial for understanding the full dimension of a matrix. Columns help provide a clear picture of how a matrix is organized.In the matrix:\[ \begin{bmatrix} -1 & 5 & 4 & 0 \ 0 & 2 & 11 & 3 \end{bmatrix} \]Each column consists of elements from each row at the same position. Here:

• The first column contains: -1, 0.
• The second column contains: 5, 2.
• The third column contains: 4, 11.
• The fourth column contains: 0, 3.

By counting these vertical groups, we determine that there are 4 columns. Columns help us understand the second part of a matrix's dimension (rows x columns), showing how many sets of numbers run vertically.

Matrix Elements Explained

Matrix elements are the individual numbers that make up the entire matrix. Each element occupies a unique position within the structure defined by its row and column. Knowing how to identify and refer to the elements helps in various matrix operations and analyses.For instance, in the matrix:\[ \begin{bmatrix} -1 & 5 & 4 & 0 \ 0 & 2 & 11 & 3 \end{bmatrix} \]

• The element at the first row and first column is -1.
• The element at the second row and second column is 2.
• The third row (even though there isn’t one here) would follow the same identification method.

Understanding elements assists in not only explaining the structure but also performing calculations like matrix addition, subtraction, and multiplication. The collection of all elements paints the complete picture of the matrix, influencing its overall properties and behavior.