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Chapter 10: Problem 7

State the dimension of the matrix. $$\left[\begin{array}{l} 12 \\ 35 \end{array}\right]$$

Short Answer

Expert verified
The matrix has a dimension of \( 2 \times 1 \).

Step by step solution

01

Identify the Number of Rows

Start by counting how many rows the matrix has. In this case, the matrix:\[\begin{array}{l}12 \35\end{array}\]contains 2 distinct rows.
02

Identify the Number of Columns

Next, count the number of columns. Each row in the matrix contains only one element, indicating that the matrix has 1 column.
03

Write the Dimension

Using the counts from Steps 1 and 2, express the dimension of the matrix in the form \( m \times n \), where \( m \) is the number of rows and \( n \) is the number of columns. Thus, the dimension is \( 2 \times 1 \).

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Key Concepts

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

Understanding Rows and Columns in Matrices
When we look at a matrix, it's essential to know both the number of rows and the number of columns. These two elements define the size and shape of a matrix. Rows run horizontally and are responsible for putting elements in a particular sequence across the matrix. Columns, on the other hand, run vertically, stacking elements one on top of the other.
  • **Rows:** These are the horizontal lines across the matrix. Every individual list of numbers from left to right forms a row.
  • **Columns:** These are the vertical lists down the matrix. If you look at just the top to bottom path, it is a column.
For instance, in a matrix that looks like:\[\begin{bmatrix}1 & 2 \3 & 4 \5 & 6 \\end{bmatrix}\]This matrix has 3 rows and 2 columns. Recognizing these row and column arrangements turns out to be important when performing operations such as addition and multiplication, as these require compatible dimensions.
Understanding Matrix Notation
Matrix notation helps in understanding and communicating about matrices easily. When representing matrices, we use square brackets or parentheses to list numbers in a two-dimensional form. The numbers inside these brackets are referred to as elements or entries of the matrix.
Consider this matrix:\[\begin{bmatrix} 12 \35 \end{bmatrix}\]It uses brackets to clearly define the matrix boundary, separating elements into rows and columns. In mathematics, the placement of each number is precise, aligning perfectly under notation, ensuring everyone reads and understands the matrix the same way. Matrix notation keeps our numbers organized and makes calculations possible.
Understanding Matrix Dimension Notation
Matrix dimension notation refers to writing down the size of a matrix by identifying both its rows and columns. It gives a quick sense of the dataset's structure without needing to look at the entire matrix.
The \( m \times n \)notation:
  • **\( m \):** Represents the number of rows in the matrix.
  • **\( n \):** Stands for the number of columns.
For example, a matrix with 4 rows and 2 columns has a dimension notation of \( 4 \times 2 \).This notation is crucial for various operations in linear algebra, such as multiplication, where the number of columns in the first matrix must align with the number of rows in the second matrix for multiplication to be possible. Paying attention to dimension notation ensures accuracy in your calculations and results.

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Most popular questions from this chapter

Use Cramer's Rule to solve the system. $$\left\\{\begin{array}{l} 10 x-17 y=21 \\ 20 x-31 y=39 \end{array}\right.$$

Use Cramer's Rule to solve the system. $$\left\\{\begin{aligned} 3 y+5 z &=4 \\ 2 x &-z=10 \\ 4 x+7 y &=0 \end{aligned}\right.$$

Use a graphing calculator to graph the solution of the system of inequalities. Find the coordinates of all vertices, rounded to one decimal place. $$\left\\{\begin{array}{l} y \geq x-3 \\ y \geq-2 x+6 \\ y \leq 8 \end{array}\right.$$

Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{aligned} x+y &>12 \\ y &<\frac{1}{2} x-6 \\ 3 x+y &<6 \end{aligned}\right.$$

Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{aligned} x^{2}+y^{2} &<9 \\ x+y &>0 \\ x & \leq 0 \end{aligned}\right.$$
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