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Chapter 9: Problem 52

Find each matrix product when possible. $$\left[\begin{array}{rr} -1 & 5 \\ 7 & 0 \end{array}\right]\left[\begin{array}{l} 6 \\ 2 \end{array}\right]$$

Short Answer

Expert verified
\(\begin{bmatrix} 4 \ 42 \end{bmatrix} \)

Step by step solution

01

Understand the matrices

The first matrix is a 2x2 matrix: \(\begin{bmatrix} -1 & 5 \ 7 & 0 \end{bmatrix}\). The second matrix is a 2x1 column vector: \(\begin{bmatrix} 6 \ 2 \end{bmatrix} \).
02

Check the dimensions

To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Here, the first matrix has 2 columns, and the second matrix has 2 rows, so the multiplication is possible.
03

Perform the matrix multiplication

Multiply the corresponding elements and sum them up: First row of the first matrix with the column vector:\[ (-1 \times 6) + (5 \times 2) = -6 + 10 = 4 \]Second row of the first matrix with the column vector:\[ (7 \times 6) + (0 \times 2) = 42 + 0 = 42 \]
04

Write the resulting matrix

The result of the multiplication is the new matrix formed:\(\begin{bmatrix} 4 \ 42 \end{bmatrix} \).

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Key Concepts

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

Matrix Dimensions
Matrix dimensions describe the size and shape of a matrix. Dimensions are written as rows x columns.
This is crucial because matrix multiplication requires specific dimensions.
When writing dimensions, always start with rows and then columns.
For example, a matrix with 2 rows and 3 columns is a 2x3 matrix.
In our exercise, the first matrix is 2x2, and the second matrix is 2x1.
Matrix Product
The matrix product, or matrix multiplication, involves multiplying matrices together.
To multiply matrices, the number of columns in the first matrix must match the number of rows in the second matrix.
In our example, the first matrix \(\begin{bmatrix} -1 & 5 ewline 7 & 0 \end{bmatrix}\) has 2 columns and the second matrix \(\begin{bmatrix} 6 ewline 2 \end{bmatrix}\) has 2 rows.
Each element of the resulting matrix is calculated by taking the dot product of a row from the first matrix with a column from the second matrix.
If any dimension criteria are not met, the matrix multiplication is not possible.
Matrix Operations
Matrix operations include addition, subtraction, and multiplication of matrices.
Multiplication, as shown in our exercise, is a bit tricky but follow these steps:
  • Identify both matrices' dimensions.
  • Ensure the number of columns in the first matrix matches the number of rows in the second.
  • Multiply each element from a row in the first matrix with the corresponding element from a column in the second matrix, then sum the results.

Using our matrices from the exercise, the steps were:
1. Multiply the first row of the first matrix by the column of the second matrix:
\((-1 脳 6) + (5 脳 2) = -6 + 10 = 4\)
2. Multiply the second row of the first matrix by the column of the second matrix:
\((7 脳 6) + (0 脳 2) = 42 + 0 = 42\)
3. Form the resulting matrix from these products, resulting in \(\begin{bmatrix} 4 ewline 42 \end{bmatrix}\).

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Most popular questions from this chapter

Solve each problem. Tire Sales The number of automobile tire sales is dependent on several variables. In one study the relationship among annual tire sales \(S\) (in thousands of dollars), automobile registrations \(R\) (in millions), and personal disposable income \(I\) (in millions of dollars) was investigated. The results for three years are given in the table. To describe the relationship among these variables, we can use the equation $$ S=a+b R+c l $$ where the coefficients \(a, b,\) and \(c\) are constants that must be determined before the equation can be used. (Source: Jarrett, J., Business Forecasting Methods, Basil Blackwell, Ltd.) (a) Substitute the values for \(S, R,\) and \(I\) for each year from the table into the equation \(S=a+b R+c I,\) and obtain three linear equations involving \(a, b,\) and \(c\) (b) Use a graphing calculator to solve this linear system for \(a, b,\) and \(c .\) Use matrix inverse methods. (c) Write the equation for \(S\) using these values for the coefficients. (d) If \(R=117.6\) and \(I=310.73,\) predict \(S .\) (The actual value for \(S\) was \(11,314 .\) ) (e) If \(R=143.8 \text { and } I=829.06, \text { predict } S . \text { (The actual value for } S \text { was } 18,481 .)\) $$\begin{array}{|c|c|c|} \hline S & R & I \\ \hline 10,170 & 112.9 & 307.5 \\\ \hline 15,305 & 132.9 & 621.63 \\ \hline 21,289 & 155.2 & 1937.13 \\\ \hline \end{array}$$

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to determine the solution set. $$\begin{aligned} x+y+z &=4 \\ 2 x-y+3 z &=4 \\ 4 x+2 y-z &=-15 \end{aligned}$$

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to determine the solution set. $$\begin{array}{c} 12 x+8 y=3 \\ 1.5 x+y=0.9 \end{array}$$

For the following system, \(D=-43, D_{x}=-43, D_{y}=0,\) and \(D_{z}=43 .\) What is the solution set of the system? $$ \begin{aligned} x+3 y-6 z &=7 \\ 2 x-y+z &=1 \\ x+2 y+2 z &=-1 \end{aligned} $$

Use the determinant theorems to find the value of each determinant. $$\left|\begin{array}{rrrr} 4 & 0 & 0 & 2 \\ -1 & 0 & 3 & 0 \\ 2 & 4 & 0 & 1 \\ 0 & 0 & 1 & 2 \end{array}\right|$$
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