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Chapter 9: Problem 3

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

Short Answer

Expert verified
The matrix has dimensions 2x1.

Step by step solution

01

Understand Matrix Dimensions

Matrix dimensions are given by the number of rows and columns. It's represented in the form \(m \times n\), where \(m\) is the number of rows and \(n\) is the number of columns.
02

Count the Rows

In the given matrix, count the number of horizontal lines. There are two values stacked vertically: \(12\) and \(35\). Therefore, the matrix has 2 rows.
03

Count the Columns

In the given matrix, each row has only one element, which means there is just a single vertical arrangement of numbers. Therefore, the matrix has 1 column.
04

Write the Dimensions

Based on the counts: 2 rows and 1 column, the dimensions of the matrix are represented as \(2 \times 1\).

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Key Concepts

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

Matrix Rows
When exploring a matrix, understanding the concept of rows is crucial. Rows in a matrix are the horizontal lines of elements. Each row contains a set of values aligned from left to right.
So, when you are asked to determine the number of rows in a matrix, picture looking across the matrix from top to bottom to count these horizontal lines.
In our example, the matrix appears as a vertical stack due to its dimensions. However, it still contains horizontal lines. Here, we have:
  • First row with element 12.
  • Second row with element 35.
Thus, this matrix has 2 rows. For any matrix, the number of rows is a fundamental aspect of its dimension.
Matrix Columns
Columns in a matrix are quite different. They are the vertical blocks that contain elements, running from top to bottom. Each column represents a vertical arrangement of numbers. To identify the number of columns in a matrix, simply count these vertical stacks.
Now, examine our example matrix. Each horizontal row contains exactly one number, aligning vertically. This indicates there鈥檚 only one column.
  • Only one column present, containing both 12 (first row) and 35 (second row).
This means that the matrix has 1 column. The significance of counting columns correctly is that it helps determine the matrix's dimensional structure. Here, the structuring contributes to the matrix's 2x1 dimension.
Matrix Representation
Matrix representation is a method to organize and present data systematically. This structure makes it easier to understand and interpret mathematical problems.
In general, a matrix is depicted in a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Each element has a specific position identified by its row and column number.
For example, any matrix can be expressed as a grid or a square bracket setup:
  • The numbers inside represent elements.
  • Position in the grid defines its row and column index.
For our matrix, it looks like this: \[ \begin{bmatrix} 12 \ 35 \end{bmatrix} \]. This makes it easy to see it has two rows, and one column. This clear presentation assists in performing matrix operations or solving mathematical equations.

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Most popular questions from this chapter

Express the function (or rule) in words. $$k(x)=\sqrt{x+2}$$

Let $$A=\left[\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right]$$ Calculate \(A^{2}, A^{3}, A^{4}, \ldots\) until you detect a pattern. Write a general formula for \(A^{n}\).

Find the partial fraction decomposition of the rational function. $$\frac{-10 x^{2}+27 x-14}{(x-1)^{3}(x+2)}$$

Use a calculator that can perform matrix operations to solve the system, as in Example 7. $$\left\\{\begin{array}{l} x+y+z+w=15 \\ x-y+z-w=5 \\ x+2 y+3 z+4 w=26 \\ x-2 y+3 z-4 w=2 \end{array}\right.$$

Solve for \(x\) $$\left|\begin{array}{lll} 1 & 0 & x \\ x^{2} & 1 & 0 \\ x & 0 & 1 \end{array}\right|=0$$
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