Problem 1 - Algebra 2

Question

State the dimensions of each matrix. \(\left[\begin{array}{rrr}{4} & {-2} & {2} \\ {1} & {4} & {1} \\ {0} & {5} & {-7}\end{array}\right]\)

Step-by-Step Solution

Answer

The matrix has 3 rows and 3 columns. Therefore, its dimensions are 3x3.

Step 1: Identify the Rows and Columns

Count the number of rows and the number of columns in the matrix. Rows are horizontal arrays of numbers, and columns are vertical arrays of numbers.

Step 2: Set up the problem

Write out the given matrices or vectors in standard form.

Step 3: Perform the matrix operations

Execute the required operations, showing key intermediate steps.

Step 4: Analyze the result

Interpret the result in terms of the original problem.

Step 5: State the conclusion

Clearly state the final answer.

Step 6: Conclude with the answer

Therefore, its dimensions are 3x3.

Key Concepts

Matrices in Algebra

Understanding matrices is essential in algebra, as they are a compact way to represent and work with sets of equations or data. Matrix algebra is a significant field in mathematics, with applications in various areas such as statistics, engineering, physics, computer science, and economics.

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The individual items in a matrix are called elements or entries. These elements can represent data points, coefficients in a system of linear equations, or any other values that can be organized in such a layout.

The importance of matrices doesn't just lie in their ability to hold multiple values in an organized manner. They also facilitate operations like addition, subtraction, multiplication, and finding the inverse of a system, which in turn helps in solving more complex equations.

Rows and Columns in Matrices

When learning about matrices, knowing how to distinguish between rows and columns is fundamental. Rows run horizontally across the matrix and can be seen as a list of elements that are side by side on the same line. In contrast, columns run vertically and stack elements from top to bottom.

To visualize this, imagine rows as the lines in a text and columns as the list of items in a grocery list. Each row or column can be thought of as a vector, which is a significant concept when it comes to understanding linear algebra and vector spaces in higher mathematics.

Matrix Size Identification

Identifying matrix size is a straightforward process that involves counting the number of rows and columns. The size of a matrix is written as 'rows by columns', commonly denoted as 'm x n', where 'm' is the number of rows and 'n' is the number of columns.

For example, looking at the matrix from the original exercise, we count three horizontal arrays, which means there are three rows. Vertically, there are also three arrays of numbers, or three columns. This would make the matrix size '3 x 3', a square matrix since the number of rows and columns is equal.

The size of a matrix is key to determining the possible operations one can perform on it, such as matrix multiplication or finding its determinant. It is also useful in establishing if two matrices are conformable for certain operations; for instance, you can only multiply two matrices if the number of columns in the first matrix matches the number of rows in the second.