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Chapter 9: Problem 12

Determine the size of each matrix. \(\left[\begin{array}{ll}x & y \\ z & w\end{array}\right]\)

Short Answer

Expert verified
The matrix is 2 x 2.

Step by step solution

01

Identify Rows

Observe the matrix and count the number of horizontal arrays. This matrix has 2 rows.
02

Identify Columns

Count the number of vertical arrays in the matrix. This matrix has 2 columns.
03

Combine Rows and Columns

Combine the number of rows and columns in the form rows x columns. In this case, the matrix has 2 rows and 2 columns, so it is a 2 x 2 matrix.

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Key Concepts

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

Understanding Matrix Rows
In a matrix, rows are the horizontal lines of elements. You can think of them like rows of seats in a theater. Each row contains a certain number of elements, lined up side by side. For example, in the matrix:
\(\begin{array}{ll} x & y \ z & w \ \end{array}\)
We have two rows:
- The first row contains the elements \ x \ and \ y \
- The second row contains the elements \ z \ and \ w \
Counting the rows in a matrix helps determine one part of its size.
Understanding Matrix Columns
Columns in a matrix are the vertical lines of elements. Think of them as columns of a building鈥檚 facade. Each column runs from top to bottom, containing elements arranged one below the other. In the same example matrix:
\(\begin{array}{ll} x & y \ z & w \ \end{array}\)
We have two columns:
- The first column contains the elements \ x \ and \ z \
- The second column contains the elements \ y \ and \ w \
Counting the columns provides the other part of determining the matrix's size.
What is a 2x2 Matrix?
A matrix is described by its dimensions, written as \ rows \ x \ columns \. For a given matrix with 2 rows and 2 columns:
\(\begin{array}{ll} x & y \ z & w \ \end{array}\)
We observe:
- There are 2 rows (horizontal lines).
- There are 2 columns (vertical lines).
This makes it a 2x2 matrix. Matrices can vary in size, but understanding how to count rows and columns is key to identifying and working with them correctly. Always remember, a matrix's size is defined by the number of its rows and columns.

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Most popular questions from this chapter

Write each matrix equation as a system of equations and solve the system by the method of your choice. $$\left[\begin{array}{lll}1 & 1 & 1 \\\0 & 1 & 1 \\\0 & 0 & 1\end{array}\right]\left[\begin{array}{l}x \\\y \\ z\end{array}\right]=\left[\begin{array}{l}4 \\\5 \\\6\end{array}\right]$$

The equation of a line through two points can be expressed as an equation involving a determinant. Show that the following equation is equivalent to the equation of the line through \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\). $$\left|\begin{array}{lll} x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \end{array}\right|=0$$

The following exercises investigate some of the properties of determinants. For these exercises let \(M=\left[\begin{array}{ll}3 & 2 \\ 5 & 4\end{array}\right]\) and \(N=\left[\begin{array}{ll}2 & 7 \\ 1 & 5\end{array}\right]\). Prove that the determinant of a product of two \(2 \times 2\) matrices is equal to the product of their determinants.

Find each product if possible. $$\left[\begin{array}{lll}-1 & 0 & 3\end{array}\right]\left[\begin{array}{r}-5 \\\1 \\\4\end{array}\right]$$

Let \(y=-x^{4}-3 x^{3}+x^{2}-9 x+8 .\) Does \(y\) approach \(\infty\) or \(-\infty\) as \(x\) approaches \(\infty ?\)
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