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Chapter 8: Problem 71

Describe how to perform scalar multiplication. Provide an example with your description.

Short Answer

Expert verified
Scalar multiplication means multiplying every element in a matrix by a scalar. So if we multiply a 2x2 matrix, represented by \[ \left[ \begin{array}{cc} 1 & 2 \ 3 & 4 \end{array} \right] \], by 2, the resulting matrix will be \[ \left[ \begin{array}{cc} 2 & 4 \ 6 & 8 \end{array} \right] \].

Step by step solution

01

Defining scalar and matrix

To begin with, choose a scalar (for instance 2) and a matrix. As an example, let's work with the matrix A = \[ \left[ \begin{array}{cc} 1 & 2 \ 3 & 4 \end{array} \right] \]. This is a 2x2 matrix.
02

Performing the Scalar Multiplication

Multiply every element in the matrix by the scalar. This means that the new matrix A' where A' = 2A becomes: A' = \[ \left[ \begin{array}{cc} 2*1 & 2*2 \ 2*3 & 2*4 \end{array} \right] \].
03

Result

The result of the multiplication is the matrix A' = \[ \left[ \begin{array}{cc} 2 & 4 \ 6 & 8 \end{array} \right] \]. After multiplying every element in the original matrix, the new matrix is twice the size.

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Key Concepts

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

Matrix Operations
Understanding matrix operations is fundamental when delving into topics such as linear algebra and precalculus. Scalar multiplication, one of the essential operations, is pretty straightforward. It involves taking a scalar, which is a single number, and multiplying it with every element within a matrix.

To put this into context, imagine you have a square or rectangle filled with numbers arranged in rows and columns. This is your matrix. When you perform scalar multiplication, you're essentially 'scaling' the matrix by the given scalar, similar to how you might zoom in or out of a picture, increasing or decreasing the value of each pixel, respectively.

Here's a simpler analogy: if you bought 2 bags of oranges and each bag had 3 oranges, then you have a total of 2 times 3, equaling 6 oranges. Scalar multiplication follows the same principle with each element of the matrix being 'multiplied' by 2 in the example provided.
Precalculus
Precalculus serves as the bridge between algebra and calculus, providing the foundational skills necessary to comprehend the more complex aspects of linear algebra and other higher-level mathematics. It introduces concepts like functions, complex numbers, and yes, matrices.

Scalar multiplication fits here as a prelude to more advanced matrix algebra that you鈥檒l encounter in calculus and beyond. Getting comfortable with this operation means you're ready to tackle matrix equations and transformations.

Grasping these concepts in precalculus will not only aid in solving matrix problems but also assist in understanding geometric transformations and vector scaling. When studying precalculus, visualizing scalar multiplication can aid learning, thematically associating the scalar's effect on the matrix to stretching or shrinking geometric figures.
Linear Algebra
Now stepping into the realm of linear algebra, scalar multiplication becomes a critical tool. This field is all about lines, planes, and vectors鈥攁 universe where matrices are the stars of the show.

In this context, scalar multiplication isn鈥檛 merely about changing the size of a matrix; it鈥檚 about transforming vectors and understanding their behavior under certain operations. For instance, if you have a vector representing a force or speed, multiplying it by a scalar will increase or decrease its magnitude, essentially accelerating or decelerating the force or speed.

The beauty of linear algebra lies in how these concepts apply to real-world problems like physics simulations, computer graphics, and data analysis. Every time you slide a bar to adjust the brightness of a picture on your smartphone, there's a matrix operation happening in the background with some scalar multiplication involved.

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Most popular questions from this chapter

Find \(A^{-1}\) by forming \([A | I]\) and then using row operations to obtain \([I | B],\) where \(A^{-1}=[B] .\) Check that \(A A^{-1}=I\) and \(A^{-1} A=I\) $$A=\left[\begin{array}{rrr} 1 & -1 & 1 \\ 0 & 2 & -1 \\ 2 & 3 & 0 \end{array}\right]$$

Use the coding matrix $$\begin{aligned}&A=\left[\begin{array}{rrr}1 & -1 & 0 \\\3 & 0 & 2 \\ -1 & 0 & -1\end{array}\right] \text { and its inverse }\\\&A^{-1}=\left[\begin{array}{rrr} 0 & 1 & 2 \\\\-1 & 1 & 2 \\\0 & -1 & -3\end{array}\right] \text { to write a cryptogram for each } \end{aligned}$$ message. Check your result by decoding the cryptogram. $$\begin{aligned}&\begin{array}{ccccccccc}\mathrm{S} & \mathrm{T} & \mathrm{A} & \mathrm{Y} & _- & \mathrm{W} & \mathrm{E} & \mathrm{L} & \mathrm{L} \\ 19 & 20 & 1 & 25 & 0 & 23 & 5 & 12 & 12\end{array}\\\&\text { Use }\left[\begin{array}{rrr} 19 & 25 & 5 \\\20 & 0 & 12 \\\1 & 23 & 12\end{array}\right]\end{aligned}$$

It's January \(1,\) and you've written down your major goal for the year. You do not want those closest to you to see what you've written in case you do not accomplish your objective. Consequently, you decide to use a coding matrix to encode your goal. Explain how this can be accomplished.

Explain how to write a linear system of three equations in three variables as a matrix equation.

Use a graphing utility to find the multiplicative inverse of each matrix. Check that the displayed inverse is correct. $$\left[\begin{array}{rrrr}7 & -3 & 0 & 2 \\\\-2 & 1 & 0 & -1 \\ 4 & 0 & 1 & -2 \\\\-1 & 1 & 0 & -1\end{array}\right]$$
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