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Chapter 9: Problem 75

For each pair of matrices \(A\) and \(B,\) find \((a) A B\) and \((b) B A\). $$A=\left[\begin{array}{rr} 3 & 4 \\ -2 & 1 \end{array}\right], B=\left[\begin{array}{rr} 6 & 0 \\ 5 & -2 \end{array}\right]$$

Short Answer

Expert verified
AB = \left[\begin{array}{rr} 38 & -8 \ -7 & -2 \end{array}\right], BA = \left[\begin{array}{rr} 18 & 24 \ 19 & 18 \end{array}\right]

Step by step solution

01

Understand Matrices Multiplication Rules

When multiplying two matrices, ensure the number of columns in the first matrix (A) matches the number of rows in the second matrix (B). The general formula for matrix multiplication involves taking the dot product of rows from the first matrix with columns of the second matrix.
02

Set up the operation for AB

Matrix A is \[\left[\begin{array}{rr} 3 & 4 \ -2 & 1 \end{array}\right]\]Matrix B is \[\left[\begin{array}{rr} 6 & 0 \ 5 & -2 \end{array}\right]\]Write the product AB as \[\left[\begin{array}{rr} (3*6 + 4*5) & (3*0 + 4*-2) \ (-2*6 + 1*5) & (-2*0 + 1*-2) \end{array}\right]\]
03

Calculate elements of AB

Calculate each element of the resulting matrix: \[\left[\begin{array}{rr} (18 + 20) & (0 - 8) \ (-12 + 5) & (0 - 2) \end{array}\right] \rightarrow \left[\begin{array}{rr} 38 & -8 \ -7 & -2 \end{array}\right]\]
04

Verify the product AB

Double-check each calculation: \[\left[\begin{array}{rr} 38 & -8 \ -7 & -2 \end{array}\right]\]AB is verified.
05

Set up the operation for BA

Write the product BA as \[\left[\begin{array}{rr} (6*3 + 0*-2) & (6*4 + 0*1) \ (5*3 + -2*-2) & (5*4 + -2*1) \end{array}\right]\]
06

Calculate elements of BA

Calculate each element of the resulting matrix: \[\left[\begin{array}{rr} (18 + 0) & (24 + 0) \ (15 + 4) & (20 - 2) \end{array}\right] \rightarrow \left[\begin{array}{rr} 18 & 24 \ 19 & 18 \end{array}\right]\]
07

Verify the product BA

Double-check each calculation: \[\left[\begin{array}{rr} 18 & 24 \ 19 & 18 \end{array}\right]\]BA is verified.

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Key Concepts

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

dot product
The dot product is essentially a way to multiply two vectors to get a scalar. In the context of matrix multiplication, it is used to calculate each element in the resulting matrix.

When multiplying matrices, we take the dot product of corresponding rows and columns from the two matrices. For example, in the first element of matrix AB, we compute the dot product of the first row of matrix A \[ \begin{bmatrix} 3 & 4 \ \end{bmatrix} \] and the first column of matrix B \[ \begin{bmatrix} 6 \ 5 \end{bmatrix} \]. The calculation is:
\[ 3 \times 6 + 4 \times 5 = 18 + 20 = 38. \]
Using the dot product, we can determine all the elements in the resulting multiplication matrix AB.
matrix operations
Matrix operations include addition, subtraction, and importantly, multiplication. Matrix multiplication is more complex as it requires the dot product.

First, ensure the two matrices can be multiplied by checking the dimensions. For example, matrix A with dimensions 2x2 can be multiplied with matrix B with dimensions 2x2. The inner dimensions must match (number of columns in A must equal number of rows in B).

The resulting matrix C will have the dimensions from the outer dimensions of A and B. So, multiplying a 2x2 matrix with another 2x2 matrix results in a 2x2 matrix.
element calculation
When calculating elements in the resulting matrix after multiplication, utilize the elements of the matrices in a step-by-step manner.

Each element in the resulting matrix is derived from the dot product of a row from the first matrix with a column of the second matrix. Here鈥檚 a small breakdown using matrix A and matrix B:
  • First element: First row of A with first column of B: \[ 3 \times 6 + 4 \times 5 = 38. \]
  • Second element: First row of A with the second column of B: \[ 3 \times 0 + 4 \times -2 = -8. \]

This way, each element of the resultant matrix AB and BA can be individually calculated and verified to ensure accuracy.
By meticulously calculating each element, matrix multiplication becomes clear and manageable.

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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 the determinant theorems to find the value of each determinant. $$\left|\begin{array}{rrrr} 5 & 1 & 4 & 2 \\ 4 & -3 & 7 & -4 \\ 5 & 8 & -3 & 6 \\ 9 & 9 & 0 & 8 \end{array}\right|$$

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to determine the solution set. $$\begin{aligned} &4 x-y=0\\\ &2 x+3 y=14 \end{aligned}$$

Gasoline Revenues The manufacturing process requires that oil refineries manufacture at least 2 gal of gasoline for each gallon of fuel oil. To meet the winter demand for fuel oil, at least 3 million gal per day must be produced. The demand for gasoline is no more than 6.4 million gal per day. If the price of gasoline is \(\$ 2.90\) per gal and the price of fuel oil is \(\$ 2.50\) per gal, how much of each should be produced to maximize revenue?

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