Problem 54 - Algebra 2

Question

State the dimensions of each matrix. $$ \left[\begin{array}{rr}{8} & {6} \\ {5} & {2} \\ {-4} & {-1}\end{array}\right] $$

Step-by-Step Solution

Answer

The matrix has dimensions 3x2.

Step 1: Identifying Matrix Type

A matrix is composed of rows and columns. The given matrix is written in a rectangular array form, surrounded by brackets. The first step in identifying the type of matrix is to count the number of rows and columns.

Step 2: Counting Rows

Look at the matrix to identify horizontal segments. Counting these segments will give you the total number of rows. For this matrix, the rows are \([8, 6]\), \([5, 2]\), and \([-4, -1]\). So, there are 3 rows.

Step 3: Counting Columns

Now, identify the vertical segments in the matrix, each representing a column. By looking at the matrix, each row has exactly 2 numbers aligned in a vertical manner. This indicates there are 2 columns.

Step 4: Writing the Dimensions

In matrix notation, dimensions are written as 'rows x columns'. With 3 rows and 2 columns identified, the matrix has dimensions \(3 \times 2\).

Key Concepts

Matrix Operations

Understanding matrix operations is essential for working with matrices effectively. These operations include addition, subtraction, scalar multiplication, and multiplication of matrices.
Matrix addition and subtraction require the matrices to have the same dimensions. This allows each element in one matrix to be added or subtracted from the corresponding element in the other matrix.
Scalar multiplication involves multiplying every element of a matrix by a single number, called a scalar.

• Addition and Subtraction: Combine matrices of identical dimensions element-wise.
• Scalar Multiplication: Multiply each element by the scalar value.

Matrix multiplication, however, has a different requirement. To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. This results in a new matrix where each element is a product of rows and columns from the original matrices.
These operations are fundamental steps in more complex matrix manipulations used in various applications, like 3D graphics and solving linear equations.

Rectangular Array

A matrix is graphically represented as a rectangular array of numbers, symbols, or expressions arranged in rows and columns. This structure helps in organizing complex data more straightforwardly.
Each element in a matrix can be referred to by its position in the rows and columns. For instance, in a 3x2 matrix, the element in the second row and first column can be noted as element (2,1).

• Row: A horizontal line of elements in the matrix.
• Column: A vertical line of elements in the matrix.
• Rectangular Shape: Formed by rows and columns meeting at right angles.

This rectangular array format allows for easy computation and dataset manipulation, providing a foundation for performing matrix operations and solving real-world problems efficiently.

Rows and Columns

Rows and columns are the fundamental components of a matrix. In any matrix, rows are the horizontal lines of elements, and columns are the vertical lines. This distinction is crucial for understanding and utilizing matrix dimensions.
Rows and columns define the size of the matrix. The number of rows tells us how many horizontal lines are present, while columns indicate the number of vertical lines. A matrix is described by its dimensions in "rows x columns" format.

• Rows: Represent horizontal arrangement; counted first in matrix dimensions.
• Columns: Represent vertical arrangement; counted second in dimensions.

Recognizing the number of rows and columns not only helps in determining dimensions but also is key to performing various matrix operations such as addition and multiplication.
Understanding these components lets you manipulate matrices to solve algebraic equations, make transformations, and more, in fields like computer science, physics, and engineering.