Problem 1 - 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. $$\left[\begin{array}{rr} -3 & 6 \\ 7 & -4 \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix is 2x2 and is a square matrix.

Step 1: Understand the Matrix Structure

A matrix is an array of numbers arranged in rows and columns. Each entry in the matrix is referred to as an element.

Step 2: Define the Matrix's Dimensions

To find the dimension of a matrix, count the number of rows and then the number of columns. The dimension is expressed as rows x columns.

Step 3: Count Rows and Columns

The given matrix is \( \left[\begin{array}{rr} -3 & 6 \ 7 & -4 \end{array}\right] \). It has 2 rows and 2 columns.

Step 4: Identify the Type of Matrix

A square matrix has the same number of rows and columns. As the given matrix has 2 rows and 2 columns, it is a square matrix. It is not a column matrix (more rows than columns) or a row matrix (more columns than rows).

Key Concepts

Square Matrix

A square matrix is a special type of matrix where the number of rows is equal to the number of columns. It's a balanced and symmetrical shape in the realm of matrices.
These matrices play a pivotal role in both theoretical and applied mathematics. They are often used in solving systems of linear equations and have properties that make complex operations feasible.

Key characteristics of square matrices include:

• Symmetry: Because the dimensions (rows and columns) are equal, calculations involving square matrices often simplify beautifully.
• Determinants: Only square matrices have a determinant, which is a value used in many calculations, including solving linear systems and finding inverses.
• Trace: The trace is the sum of the elements on the main diagonal of a square matrix.

For example, the matrix \[\begin{bmatrix}-3 & 6 \7 & -4 \\end{bmatrix}\] is a square matrix because it contains exactly 2 rows and 2 columns, making it extremely useful for various calculations that rely on this balance.

Row Matrix

A row matrix is defined as a matrix containing a single row. This means there is only one horizontal sequence of elements, and potentially many columns.
They are used quite frequently when dealing with data or calculations where individual vectors are being analyzed or manipulated.

Characteristics of a row matrix include:

• It has only one row and any number of columns.
• It is often represented as a 1 × n matrix where 'n' is the number of columns.
• Because there is only a single data line (or row), they can simplify operations like addition or scalar multiplication with other matrices of similar or compatible structures.

For instance, a matrix like \[\begin{bmatrix}2 & 1 & 4 & 5\end{bmatrix}\] is a row matrix. It has just one row, but four columns, making it a 1 × 4 matrix. This structure contrasts greatly with square matrices in application but is quite specialized in its own right.

Column Matrix

Column matrices are essentially the vertical cousins to row matrices. A column matrix contains a single column but can have numerous rows.
They are vital in the study of vector spaces and arise frequently when solving equations in one variable or managing single-dimensional data sets.

Attributes of a column matrix include:

• It consists of one column and multiple rows.
• It is often denoted as an m × 1 matrix where 'm' stands for the number of rows.
• Column matrices are effectively used to represent vectors in higher-dimensional spaces, adding a vertical layer to calculations.

An example of a column matrix is:\[\begin{bmatrix}3 \6 \9 \12\end{bmatrix}\]This matrix is a column matrix due to its 4 rows and 1 single column, classified as a 4 × 1 matrix. This form makes them suited for operations needing linear data to be arranged vertically.