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Chapter 9: Problem 14

Concept Check Find the dimension of each matrix. Identify any square, column, or row matrices. $$\left[\begin{array}{rrr} -9 & 6 & 2 \\ 4 & 1 & 8 \end{array}\right]$$

Short Answer

Expert verified
The dimension is 2x3. It is not a square, row, or column matrix.

Step by step solution

01

Identify the dimensions of the matrix

Count the number of rows and columns in the matrix. The given matrix is \[ \begin{array}{rrr} -9 & 6 & 2 \ 4 & 1 & 8 \ \ \ \end{array} \] It has 2 rows and 3 columns. Therefore, the dimension of the matrix is 2x3.
02

Determine the type of matrix

Check the dimensions and see if the matrix is a square matrix, column matrix, or row matrix. A square matrix has the same number of rows and columns, a column matrix has only one column, and a row matrix has only one row. Since this matrix has 2 rows and 3 columns, it is neither square, row, nor column matrix.

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Key Concepts

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

Matrix Types
Matrices are an important part of precalculus. They come in different types depending on their dimensions and structure. Here are the main types of matrices:
  • Square Matrix: A matrix with the same number of rows and columns. For example, a 3x3 matrix.
  • Row Matrix: A matrix with only one row. An example would be a 1x3 matrix.
  • Column Matrix: A matrix with only one column. For instance, a 3x1 matrix.
It's essential to identify the type of matrix as it helps in applying the correct mathematical operations.
Rows and Columns
When working with matrices, counting rows and columns is crucial. A matrix is represented in the form \[\begin{array}{rrr} a & b & c \ d & e & f \ \end{array}\]. Each horizontal line of elements is called a row, and each vertical line of elements is called a column.

In the given matrix \[\begin{array}{rrr}-9 & 6 & 2 \ 4 & 1 & 8 \end{array}\], we see:\br>
  • It has 2 rows.
  • It has 3 columns.
Therefore, this matrix has dimensions of 2x3 (2 by 3). This simple counting will help you identify matrix dimensions quickly.
Square Matrix
A square matrix is a special type of matrix where the number of rows equals the number of columns. For example, the matrix \[\begin{array}{rrr}1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{array}\] is a 3x3 matrix, making it square.

These matrices are significant because they have properties like a determinable determinant and an inverse under certain conditions. Not every matrix is a square matrix. In the example given \[\begin{array}{rrr}-9 & 6 & 2 \ 4 & 1 & 8 \end{array}\], this matrix is not square because its dimensions are 2x3.
Always check if the row number equals the column number to determine if a matrix is square.

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Most popular questions from this chapter

Solve each system by using the inverse of the coefficient matrix. $$\begin{aligned} &\frac{1}{2} x+\frac{1}{3} y=\frac{49}{18}\\\ &\frac{1}{2} x+2 y=\frac{4}{3} \end{aligned}$$

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to determine the solution set. $$\begin{aligned} &3 x+2 y=-4\\\ &5 x-y=2 \end{aligned}$$

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