Problem 6 - A Graphical Approach to College Algebra

Question

Find the dimension of each matrix. Identify any square, column, or row matrices. Do not use a calculator. $$\left[\begin{array}{ll} 4 & 9 \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix is a \(1 \times 2\) row matrix.

Step 1: Determine the Dimensions of the Matrix

The given matrix is \( \begin{bmatrix} 4 & 9 \end{bmatrix} \), which contains one row and two columns. Therefore, we say it has the dimensions \(1 \times 2\).

Step 2: Identify the Type of Matrix

For a matrix to be square, the number of rows must equal the number of columns. A row matrix has only one row, and a column matrix has only one column. The given matrix \( \begin{bmatrix} 4 & 9 \end{bmatrix} \) has one row and two columns, so it is a row matrix.

Key Concepts

Matrix Types

Matrices are powerful mathematical tools used to organize and manipulate data that come in different shapes and sizes. Depending on their form, matrices can be classified into several types:

• Square Matrix: A matrix with an equal number of rows and columns. For example, a matrix of size 2x2 or 3x3 is considered square.
• Row Matrix: This type of matrix has only one row but can have multiple columns, like in the given problem.
• Column Matrix: Contrary to a row matrix, this one has a single column but can contain multiple rows.

Understanding these types helps in determining specialized properties and operations that can be applied to matrices.

Square Matrix

A square matrix is one where the number of rows equals the number of columns. This symmetry gives square matrices unique properties, which often simplify calculations in algebra, such as determinant and eigenvalues.

Key characteristics of square matrices include:

• The matrix can have properties like symmetry (the matrix is equal to its transpose), diagonal dominance, or invertibility.
• Determinants are only defined for square matrices, which play a crucial role in solving systems of equations.
• Square matrices of size 1x1 are referred to as scalars, and size 2x2 are the simplest non-trivial cases often used in introductory exercises.

In our given problem, the matrix is not square because it has dimensions of 1x2, meaning it doesn't have the equal row-to-column ratio necessary to be a square matrix.

Row Matrix

A row matrix is characterized by having exactly one row and any number of columns. This means you can think of it as a single line of entries lying side by side. Row matrices are often used to represent vector-like data that is arranged horizontally.

The given matrix in the exercise, \( \begin{bmatrix} 4 & 9 \end{bmatrix} \), is a perfect example of a row matrix given its 1x2 dimension:

• It has one single row.
• The two numbers it contains, 4 and 9, are its columns.
• This makes it highly efficient for operations like dot product calculations with column vectors.

By recognizing the matrix as a row matrix, we know it utilizes horizontal data representation, distinct from a column matrix or more complex square matrix forms.