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Chapter 8: Problem 5

Determine the order of each matrix. $$[0]$$

Short Answer

Expert verified
The order of the matrix is 1x1.

Step by step solution

01

Understanding Matrix Order

The order of a matrix is defined by the number of its rows and columns. The order is written as \( m \times n \) where \( m \) is the number of rows and \( n \) is the number of columns.
02

Analyze the Given Matrix

The given matrix is \( [0] \). It has one row and one column. Thus, it is a 1x1 matrix.
03

Determine the Order

The matrix \( [0] \) has one row and one column, so the order of the matrix is \( 1 \times 1 \).

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Key Concepts

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

Rows and Columns of a Matrix
Matrices consist of numbers arranged in a rectangular grid, called rows and columns. When you hear someone talk about the 'order' of a matrix, they're referring to how many rows and columns make it up. The number of rows is listed first and the number of columns second 鈥� like an address to help you find your way around the matrix. Here鈥檚 how it works:
  • Rows are horizontal lines of numbers.
  • Columns are vertical lines of numbers.
For example, a matrix with three rows and two columns is called a 3x2 matrix. In a matrix like this, you would have three separate horizontal lists of two numbers each. This convention helps to quickly comprehend the structure and size of the matrix layout at a glance.
1x1 Matrix
A 1x1 matrix is the simplest form of a matrix you can encounter. It consists of a single element arranged neatly in one row and one column. Here鈥檚 what it looks like:
  • One number positioned alone in a grid.
Because it only has one element, a 1x1 matrix can be thought of as equivalent to a single number. However, it鈥檚 important to consider it a matrix for calculation purposes, especially when dealing with matrix operations in larger systems. This understanding helps in appreciating even a small matrix's role within broader mathematical contexts.
Matrix Notation
Matrix notation is a standardized way to express or name matrices, encapsulating their size and structure. This notation typically takes the form of a bracketed array, but it begins with listing its order. Here鈥檚 how you can understand this notation better:
  • The matrix is denoted with square brackets like \[\begin{bmatrix}a & b \c & d\end{bmatrix}\]for a 2x2 matrix with elements \(a, b, c,\) and \(d\).
  • Order is crucial. For a 3x3 matrix, it would include 3 rows and 3 columns.
  • Elements within the matrix are addressed by their position, corresponding to their row and column numbers, like \( a_{ij} \) where \( i \) is the row and \( j \) is the column.
Understanding matrix notation reinforces the way matrices are structured and manipulated, serving as a versatile tool for various applications in mathematics and data analysis.

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Most popular questions from this chapter

Apply Cramer's rule to solve each system of equations and a graphing utility to evaluate the determinants. $$\begin{array}{l} 3.1 x+1.6 y-4.8 z=-33.76 \\ 5.2 x-3.4 y+0.5 z=-36.68 \\ 0.5 x-6.4 y+11.4 z=25.96 \end{array}$$

For the system of equations $$ \begin{array}{l} 3 x+2 y=5 \\ a x-4 y=1 \end{array} $$ find \(a\) that guarantees no unique solution.

In Exercises 77 and 78 , explain the mistake that is made. Graph the inequality \(y < 2 x+1\) Solution: Graph the line \(y=2 x+1\) with a solid line. (GRAPH CAN'T COPY). since the inequality is \( < ,\) shade below. (GRAPH CAN'T COPY). This is incorrect. What mistake was made?

Why does the square matrix \(A=\left[\begin{array}{ll}2 & 3 \\ 4 & 6\end{array}\right]\) not have an inverse?

Determine whether the statements are true or false. $$\begin{array}{l}\text { If } A=\left[\begin{array}{ll}a_{11} & a_{12} \\\a_{21} & a_{22} \end{array}\right] \text { and } B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right], \text { then } \\\A B=\left[\begin{array}{ll}a_{11} b_{11} & a_{12} b_{12} \\\a_{21} b_{21} & a_{22} b_{22}\end{array}\right]\end{array}$$
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