Problem 7 - Intermediate Algebra

Question

For each matrix, determine the number of rows and the number of columns. a. \(\left[\begin{array}{ccc}4 & 6 & -1 \\ 1 & 9 & -3\end{array}\right]\) b. \(\left[\begin{array}{rrrr}1 & -2 & 3 & 1 \\ 0 & 1 & 6 & 4 \\ 0 & 0 & 1 & \frac{1}{3}\end{array}\right]\)

Step-by-Step Solution

Answer

Matrix (a) has 2 rows and 3 columns; matrix (b) has 3 rows and 4 columns.

Step 1: Identifying Rows and Columns in Matrix (a)

In matrix (a), examine the array of numbers to determine its dimensions. The matrix is given as \(\begin{bmatrix} 4 & 6 & -1 \ 1 & 9 & -3 \end{bmatrix}\). Count the horizontal lines of numbers to find the number of rows and the vertical lines of numbers to find the number of columns. This matrix has 2 rows and 3 columns.

Step 2: Identifying Rows and Columns in Matrix (b)

Now examine matrix (b), which is given as \(\begin{bmatrix} 1 & -2 & 3 & 1 \ 0 & 1 & 6 & 4 \ 0 & 0 & 1 & \frac{1}{3} \end{bmatrix}\). Count the horizontal sets of numbers to determine the number of rows and the vertical sets for the number of columns. This matrix consists of 3 rows and 4 columns.

Key Concepts

Row and Column Identification

In the context of matrices, understanding how to identify rows and columns is crucial. Rows are horizontal arrangements of elements, while columns are vertical ones. To determine the number of rows in a matrix, inspect the horizontal bands of numbers. On the other hand, to identify the number of columns, look at how many numbers are vertically stacked.
For example, in the matrix \[ \begin{bmatrix} 4 & 6 & -1 \ 1 & 9 & -3 \end{bmatrix} \] this array has 2 horizontal lines of numbers, meaning it has 2 rows. To find columns, count the vertical set of numbers; for this matrix, 3 elements make up each vertical group, leading to 3 columns.

• Rows: determined by horizontal lines of elements.
• Columns: determined by vertical lines of elements.

Matrix Notation

Matrix notation is a systematic way to express a collection of numbers, symbols, or expressions arranged in rows and columns. This kind of notation is typically represented within brackets. The number of rows and columns determines the size of a matrix, often expressed as 'm x n', where 'm' represents rows and 'n' columns. For instance, a matrix labeled as a 2 x 3 matrix indicates 2 rows and 3 columns. It is important to read matrix dimensions as row count first, followed by column count. This order helps when performing matrix operations, like addition or multiplication.

• 'm x n' notation: 'm' is the number of rows, 'n' is the number of columns.
• Brackets: used to encapsulate the matrix.

Matrices

Matrices are collections of numbers organized into a rectangular grid comprised by rows and columns. They are a fundamental concept in linear algebra and widely used in various fields such as physics, computer science, and economics. The versatility of matrices allows them to represent systems of equations, transformations in graphics, and even data structures in programming. A matrix can be described by its dimensions, i.e., the number of rows and columns, which dictate how they can interact with other matrices through operations like addition or multiplication.
Other operations, such as transposition and inversion, further utilize the setup and configurations provided by matrices' structural dimensions.

• Matrix as a rectangular array.
• Applications: equations, transformations, data structures.