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Chapter 1: Problem 4

Determine the size of the matrix. $$ (-1) $$

Short Answer

Expert verified
The size of the given matrix is 1x1

Step by step solution

01

Identify the number of rows and columns

As this matrix is a single number, it can be considered to exist in a 1x1 matrix. Therefore, the given matrix only has one row and one column.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

1x1 matrix
A 1x1 matrix is one of the simplest forms of a matrix. It features only a single element and is organized into one row and one column. Imagine a box containing only one number, like -1, as seen in the exercise. This box can be thought of as a grid, but with only one slot filled.
This makes a 1x1 matrix very easy to handle since all operations are straightforward, involving just the element itself.
  • Adding or multiplying a 1x1 matrix is like dealing with a single number.
  • It serves as a convenient way to abstract a single data value into matrix operations.
Understanding this simple concept helps in tackling more complex matrices down the line. Once you grasp a 1x1 matrix, you've made the first step into the broader world of matrix algebra.
matrix dimensions
The dimensions of any matrix refer to the number of its rows and columns. A general matrix could look like a table, where these numbers specify how many slots are horizontally (columns) and vertically (rows). You express dimensions in the form of 'rows x columns', also written as 'r x c'.
In the context of our example, the 1x1 matrix has one row and one column, hence its dimensions are 1x1. This notation succinctly encapsulates the structure of the matrix.
  • Matrix dimensions allow us to understand and use multiple data points together, structured neatly in rows and columns.
  • They are crucial for defining operations like addition, where both matrices need to be of the same dimensions.
Without knowing matrix dimensions, you can鈥檛 effectively work with matrices in algebra, so always start by checking the size.
single-element matrix
A single-element matrix, as the name suggests, is composed of just one element. This concept is pivotal because it bridges the gap between numerical data and matrix theory.
Single-element matrices can represent simple scalars but within a matrix context. This gives them special properties, allowing mathematicians to treat numbers as if they were more complex structures.
  • While they are minimalistic, their implications in transformations and numerical computations are significant.
  • This characteristic makes them useful in various applications, including physics and engineering computations.
With this understanding, single-element matrices are more than just a curiosity in mathematics鈥攖hey are foundational pieces of larger puzzles.

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Most popular questions from this chapter

Determine whether the matrix is in row-echelon form. If it is, determine whether it is also in reduced rew-echelon form. $$ \left[\begin{array}{llll} 1 & 0 & 2 & 1 \\ 0 & 1 & 3 & 4 \\ 0 & 0 & 1 & 0 \end{array}\right] $$

Find values of \(a, b,\) and \(c\) (if possible) such that the system of linear equations has (a) a unique solution, (b) no solution, and (c) infinitely many solutions. $$ \begin{aligned} x+y &=2 \\ y+z &=2 \\ x+z &=2 \\ a x+b y+c z &=0 \end{aligned} $$

Determine the value(s) of \(k\) such that the system of linear equations has the indicated number of solutions. Infinitely many solutions \(k x+y=16\) \(3 x-4 y=-64\)

Writing Consider the \(2 \times 2\) matrix \(\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\) Perform the sequence of row operations. (a) Add \((-1)\) times the second row to the first row. (b) Add 1 times the first row to the second row. (c) Add \((-1)\) times the second row to the first row. (d) Multiply the first row by \((-1)\) What happened to the original matrix? Describe, in general, how to interchange two rows of a matrix using only the second and third elementary row operations.

Find the reduced row-echelon matrix that is row-equivalent to the given matrix. $$ \left[\begin{array}{rr} 1 & 2 \\ -1 & 2 \end{array}\right] $$
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