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Chapter 9: Problem 56

Find each matrix product when possible. $$\left[\begin{array}{rrr} -9 & 2 & 1 \\ 3 & 0 & 0 \end{array}\right]\left[\begin{array}{r} \sqrt{5} \\ \sqrt{20} \\ -2 \sqrt{5} \end{array}\right]$$

Short Answer

Expert verified
The resulting matrix is [[-3 sqrt{5}],[3 sqrt{5}]]

Step by step solution

01

Verify Matrix Dimensions

Check if the number of columns in the first matrix (2x3) matches the number of rows in the second matrix (3x1). Since both have the dimension 3, the multiplication is possible.
02

Write the Formula for Matrix Multiplication

The general formula for matrix multiplication of matrices A (m x n) and B (n x p) is given by element C_{i,j} = 危 (A_{i,k} * B_{k,j}) from k=1 to n.
03

Calculate Each Element of the Resulting Matrix

Multiply each element of the rows of the first matrix by the corresponding elements of the columns of the second matrix, and sum the products. For element C_{1,1}: (-9 * sqrt{5}) + (2 * sqrt{20}) + (1 * -2 sqrt{5}) = -9 sqrt{5} + 8 sqrt{5} - 2 sqrt{5} == -3 sqrt{5} For element C_{2,1}: (3 * sqrt{5}) + (0 * sqrt{20}) + (0 * -2 sqrt{5}) = 3 sqrt{5}
04

Form the Resulting Matrix

Combine the calculated elements into the resulting matrix. Therefore, the resultant product matrix is: [ [-3 sqrt{5}], [3 sqrt{5}] ]

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Key Concepts

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

Matrix Dimensions
Understanding matrix dimensions is crucial for performing matrix multiplication. Matrices are often denoted by their dimensions as **m** x **n**, where **m** represents the number of rows and **n** represents the number of columns. For instance, if you have a matrix that has 2 rows and 3 columns, it can be referred to as a 2x3 matrix.

In matrix multiplication, the rule is that the number of columns in the first matrix must be equal to the number of rows in the second matrix. Only then is the matrix multiplication possible. For example, in the given problem:
  • Matrix A has dimensions 2x3 (2 rows and 3 columns)
  • Matrix B has dimensions 3x1 (3 rows and 1 column)
Since the number of columns in Matrix A (3) matches the number of rows in Matrix B (3), the multiplication can proceed.
Matrix Product
The matrix product is essentially the resulting matrix you get after performing matrix multiplication. The resulting matrix has dimensions derived from the dimensions of the two matrices being multiplied. Specifically, if a matrix A has dimensions **m** x **n** and matrix B has dimensions **n** x **p**, the resulting matrix will have dimensions **m** x **p**.

For the example provided:
  • Matrix A (2x3) multiplied by Matrix B (3x1) results in a matrix with dimensions 2x1.
In other words, the resulting matrix will have 2 rows (from Matrix A) and 1 column (from Matrix B). This concept is fundamental because it helps students predict the size of the matrix they will end up with after multiplication.
Matrix Element Calculation
Calculating elements in the resulting matrix involves precise arithmetic operations based on the elements of the original matrices. Each element in the product matrix, denoted as \(C_{i,j}\), is obtained by multiplying corresponding elements from the rows of the first matrix (A) and the columns of the second matrix (B), and then summing these products.

The formula for determining each element is: \[ C_{i,j} = \sum_{k=1}^{n} (A_{i,k} * B_{k,j})\] where:
  • \(C_{i,j}\) represents the element in the resulting matrix at row **i** and column **j**
  • \(A_{i,k}\) represents the element at row **i**, column **k** in matrix A
  • \(B_{k,j}\) represents the element at row **k**, column **j** in matrix B
For the given matrices:

  • For element \(C_{1,1}\): \((-9 * \sqrt{5}) + (2 * \sqrt{20}) + (1 * -2 \sqrt{5}) = -9 \sqrt{5} + 8 \sqrt{5} - 2 \sqrt{5} = -3 \sqrt{5}\)
  • For element \(C_{2,1}\): \((3 * \sqrt{5}) + (0 * \sqrt{20}) + (0 * -2 \sqrt{5}) = 3 \sqrt{5}\)
Once all elements are calculated, you compile them into the resulting matrix: \([-3 \sqrt{5}], [3 \sqrt{5}]\)

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Most popular questions from this chapter

Solve each problem. A sparkling-water distributor wants to make up 300 gal of sparkling water to sell for \(\$ 6.00\) per gallon. She wishes to mix three grades of water selling for \(\$ 9.00, \$ 3.00,\) and \(\$ 4.50\) per gallon, respectively. She must use twice as much of the S4.50 water as of the \(\$ 3.00\) water. How many gallons of each should she use?

Supply and Demand In many applications of economics, as the price of an item goes up, demand for the item goes down and supply of the item goes up. The price where supply and demand are equal is the equilibrium price, and the resulting sup. ply or demand is the equilibrium supply or equilibrium demand. Suppose the supply of a product is related to its price by the equation $$p=\frac{2}{3} q$$ where \(p\) is in dollars and \(q\) is supply in appropriate units. (Here, \(q\) stands for quantity.) Furthermore, suppose demand and price for the same product are related by $$p=-\frac{1}{3} q+18$$ where \(p\) is price and \(q\) is demand. The system formed by these two equations has solution \((18,12),\) as seen in the graph. (GRAPH CANNOT COPY) Find the demand for the electric can opener at each price. (a) \(\$ 6\) (b) \(\$ 11\) (c) \(\$ 16\)

Use a system of equations to solve each problem. Find the equation of the parabola \(y=a x^{2}+b x+c\) that passes through the points \((2,3),(-1,0),\) and \((-2,2)\)

For the system below, match each determinant in (a)-(d) with its equivalent from choices \(\mathrm{A}-\mathrm{D}\). $$ \begin{aligned} 4 x+3 y-2 z &=1 \\ 7 x-4 y+3 z &=2 \\ -2 x+y-8 z &=0 \end{aligned} $$ \(\begin{array}{lll}\text { (a) } D & \text { (b) } D_{x}\end{array}\) (c) \(D_{y}\) (d) \(D_{z}\) A. \(\left|\begin{array}{rrr}1 & 3 & -2 \\ 2 & -4 & 3 \\ 0 & 1 & -8\end{array}\right|\) B. \(\left|\begin{array}{rrr}4 & 3 & 1 \\ 7 & -4 & 2 \\\ -2 & 1 & 0\end{array}\right|\)C. \(\left|\begin{array}{rrr}4 & 1 & -2 \\ 7 & 2 & 3 \\ -2 & 0 & -8\end{array}\right|\) D. \(\left|\begin{array}{rrr}4 & 3 & -2 \\\ 7 & -4 & 3 \\ -2 & 1 & -8\end{array}\right|\)

Solve each problem. The sum of two numbers is \(47,\) and the difference between the numbers is 1. Find the numbers.
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