Chapter 7.2, Problem 47E

Solution

a.

To calculate: The product of the matrix $B^{T}A$ by using the matrix $A$ and $B$ .

  $A=\left[\begin{array}{cc}100& 60\\ 120& 70\\ 200& 120\end{array}\right]$ , $B=\left[\begin{array}{c}\$0.80\\ \$0.85\\ \$1.00\end{array}\right]$

The obtained matrix is $\left[\begin{array}{cc}\$382& \$227.5\end{array}\right]$ .

Given Information:

The matrix are,

  $A=\left[\begin{array}{cc}100& 60\\ 120& 70\\ 200& 120\end{array}\right]$ , $B=\left[\begin{array}{c}\$0.80\\ \$0.85\\ \$1.00\end{array}\right]$

Formula Used:

To find the transpose matrix, convert row to column.

Calculation:

Consider the given matrix,

  $A=\left[\begin{array}{cc}100& 60\\ 120& 70\\ 200& 120\end{array}\right]$ , $B=\left[\begin{array}{c}\$0.80\\ \$0.85\\ \$1.00\end{array}\right]$

Now, find multiplication.

  $\begin{array}{c}{B}^{T}A={\left[\begin{array}{c}\$0.80\\ \$0.85\\ \$1.00\end{array}\right]}^T\cdot \left[\begin{array}{cc}100& 60\\ 120& 70\\ 200& 120\end{array}\right]\\ =\left[\begin{array}{ccc}\$0.80& \$0.85& \$1.00\end{array}\right]\cdot \left[\begin{array}{cc}100& 60\\ 120& 70\\ 200& 120\end{array}\right]\\ =\left[\begin{array}{cc}0.8\left(100\right)+0.85\left(120\right)+200& 0.8\left(60\right)+0.85\left(70\right)+120\end{array}\right]\\ =\left[\begin{array}{cc}\$382& \$227.5\end{array}\right]\end{array}$

Hence, the obtained product is $\left[\begin{array}{cc}\$382& \$227.5\end{array}\right]$ .

b.

To determine: The explanation for the obtained result is part (a).

The total amount charges grocery store $j$ for eggs by Happy Valley farms.

Given Information:

The matrix are,

  $A=\left[\begin{array}{cc}100& 60\\ 120& 70\\ 200& 120\end{array}\right]$ , $B=\left[\begin{array}{c}\$0.80\\ \$0.85\\ \$1.00\end{array}\right]$

Explanation:

Consider the given matrix,

  $A=\left[\begin{array}{cc}100& 60\\ 120& 70\\ 200& 120\end{array}\right]$ , $B=\left[\begin{array}{c}\$0.80\\ \$0.85\\ \$1.00\end{array}\right]$

The obtained result is part (a) is $B^{T}A=\left[\begin{array}{cc}\$382& \$227.5\end{array}\right]$ .

The obtained result is explaining that the total amount charges grocery store $j$ for eggs by Happy Valley farms.

The obtained result has been explained above.