Problem 11 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine the dimension of the matrix. $$\left[\begin{array}{rr} 33 & 45 \\ -9 & 20 \end{array}\right]$$

Step-by-Step Solution

Answer

The dimension of the matrix is 2x2.

Step 1: Identifying the number of rows in the matrix

Count the number of rows in the matrix. In this case, the matrix has two rows.

Step 2: Identifying the number of columns in the matrix

Count the number of columns in the matrix. In this case, the matrix has two columns.

Step 3: Writing the dimension of the matrix

After counting the number of rows and columns, the dimension of the matrix is written as 'number of rows x number of columns'. In this case, the dimension of the matrix is 2x2.

Key Concepts

Matrix Rows

When discussing matrices, rows refer to the horizontal lines of elements. Each row contains a sequence of numbers. Identifying rows is the first step when evaluating the dimensions of a matrix. In the given example matrix, we have:

• Row 1: \(33, 45\)
• Row 2: \(-9, 20\)

For a matrix to have clear dimension notation, start by counting these rows. Here, we identify two distinct horizontal lines, meaning there are 2 rows.
Understanding rows is crucial as they help determine the dimensional labeling of each specific matrix structure.

Matrix Columns

Columns are the vertical lines in a matrix and run from top to bottom. Each column in a matrix comprises numbers stacked on top of each other. To evaluate matrix dimensions, counting columns is pivotal. For our example:

• Column 1: \(33, -9\)
• Column 2: \(45, 20\)

By aligning numbers vertically, we arrive at two vertical sets, indicating two columns. The number of columns is used alongside rows to define the overall matrix dimension.
Recognizing columns allows you to write the dimensions compactly and accurately.

2x2 Matrix

A 2x2 matrix specifically refers to a matrix configuration where there are two rows and two columns. It serves as a basic model that many students initially learn when exploring matrices. The matrix dimension is expressed in the format 'rows x columns', so our example is written as \(2 \times 2\).Key features of a 2x2 matrix include:

• Easy manipulation for operations like addition, subtraction, or multiplication.
• Often used in introductory level problems and can represent linear transformations such as rotations or scalings in 2D space.

Understanding a 2x2 matrix is fundamental for building a solid grasp of more complex matrix operations and concepts.