Problem 1 - Algebra and Trigonometry

Question

1–6 State the dimension of the matrix. $$\left[\begin{array}{cc}{2} & {7} \\ {0} & {-1} \\ {5} & {-3}\end{array}\right]$$

Step-by-Step Solution

Answer

The dimension of the matrix is 3 x 2.

Step 1: Understand Rows and Columns

A matrix is composed of rows and columns. Each horizontal line of numbers represents a row, and each vertical line represents a column. In the given matrix: \[\begin{bmatrix}2 & 7 \0 & -1 \5 & -3\end{bmatrix}\]there are multiple rows and columns.

Step 2: Count the Number of Rows

Count how many horizontal lines there are in the matrix. For the given matrix:\[\begin{bmatrix}2 & 7 \0 & -1 \5 & -3\end{bmatrix}\]There are 3 rows.

Step 3: Count the Number of Columns

Count how many vertical lines there are in the matrix. For this matrix:\[\begin{bmatrix}2 & 7 \0 & -1 \5 & -3\end{bmatrix}\]There are 2 columns.

Step 4: State the Dimension of the Matrix

The dimension of a matrix is stated as (number of rows) x (number of columns). Thus, for the matrix:\[\begin{bmatrix}2 & 7 \0 & -1 \5 & -3\end{bmatrix}\]The dimension is \(3 \times 2\).

Key Concepts

Matrix Rows

When you look at a matrix, the first thing you'll likely notice are its rows. Rows are important because they determine the matrix's dimension alongside columns. Here's how you can understand rows better:

• Rows are the horizontal arrangements of numbers in a matrix.
• Each row contains a series of numbers lined up horizontally, one after the other.
• Think of each row like a row of seats in a theater or a series of horizontal rungs on a ladder.

Consider the matrix given in our exercise:\[ \begin{bmatrix}2 & 7 \ 0 & -1 \ 5 & -3\end{bmatrix} \]This matrix has 3 rows: each can be thought of as occupying a separate horizontal path across the matrix. It's important to count them carefully, because the number of rows tells us part of the matrix's size.

Matrix Columns

Columns play a crucial role in determining the dimension of a matrix just like rows. Getting to know columns is a key skill when working with matrices:

• Columns are the vertical stacks of numbers in a matrix.
• Each column runs from top to bottom, much like the columns that hold up a temple roof.

To visualize this, revisit the matrix from our exercise:\[ \begin{bmatrix}2 & 7 \ 0 & -1 \ 5 & -3\end{bmatrix} \]The matrix contains 2 columns, each aligned vertically from top to bottom.By understanding how columns are formed, you can quickly determine one half of the matrix's overall dimension.

Matrix Size

Matrix size, an essential concept, is defined by the number of rows and columns it contains. To state the size of a matrix, we use the format (rows \(\times\) columns).

• The row number comes first, followed by a multiplication sign \(\times\).
• The column number comes second, after the multiplication sign.

For our example matrix:\[ \begin{bmatrix}2 & 7 \ 0 & -1 \ 5 & -3\end{bmatrix} \]There are 3 rows and 2 columns. Therefore, we articulate its size as a \(3 \times 2\) matrix.The size of the matrix conveys crucial information about its structure and possible operations you can perform on it.