Problem 5 - Algebra and Trigonometry

Question

1–6 State the dimension of the matrix. $$\left[\begin{array}{lll}{1} & {4} & {7}\end{array}\right]$$

Step-by-Step Solution

Answer

The dimension of the matrix is \(1 \times 3\).

Step 1: Identify Number of Rows

A matrix is defined by its rows and columns. First, count how many rows there are. In this matrix, there is only 1 row.

Step 2: Identify Number of Columns

Next, count the number of columns. In this matrix, there are 3 separate entries horizontally, making 3 columns.

Step 3: State the Dimension

The dimension of a matrix is given by the number of rows by the number of columns. Since this matrix has 1 row and 3 columns, the dimension is written as \(1 \times 3\).

Key Concepts

Matrix Definition

A matrix is a rectangular arrangement of numbers, symbols, or expressions, set out in rows and columns. Imagine it as a grid where each spot in that grid holds a specific number or symbol. Matrices are widely used in various fields such as mathematics, physics, and computer science, to perform different calculations or represent data.

• Matrices help to organize information more systematically.
• They are used in solving linear equations, transformations, and even data analysis.

To define a matrix more precisely, we use two key terms: rows and columns. This leads us to our next concepts.

Matrix Rows

In any given matrix, rows are the horizontal lines filled with elements across the matrix. You can think of rows as the different layers or levels lying horizontally. This plays a crucial role in defining the matrix's dimensions.

• The number of rows in a matrix affects its structure.
• In operations like matrix addition, the matrices must have the same number of rows to be compatible.

For instance, in the exercise's matrix emat{egin{array}{lll}{1} & {4} & {7} emat], we can see that there is just 1 row where the elements 1, 4, and 7 are laid out.
When identifying rows, always count the distinct horizontal levels present.

Matrix Columns

Columns in a matrix refer to the vertical arrangements of elements. Each column holds a series of numbers or items stacked in a vertical line. Just like rows, columns are essential in determining the matrix dimensions.

• Columns determine what can be multiplied with the matrix.
• While transposing, columns become rows and vice versa.

Looking at our example matrix emat{egin{array}{lll}{1} & {4} & {7} emat], we see 3 columns consisting of the elements 1, 4, and 7.
Counting these vertical lines and their content helps us to determine how wide our matrix is.