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Chapter 6: Problem 61

Find each matrix product if possible. $$\left[\begin{array}{rrr} 2 & 2 & -1 \\ 3 & 0 & 1 \end{array}\right]\left[\begin{array}{rr} 0 & 2 \\ -1 & 4 \\ 0 & 2 \end{array}\right]$$

Short Answer

Expert verified
The matrix product is \( \left[\begin{array}{cc}-2 & 10 \\ 0 & 8 \end{array}\right] \).

Step by step solution

01

Verify Dimensions for Matrix Multiplication

We have two matrices to multiply: Matrix A with dimensions 2x3, and Matrix B with dimensions 3x2. For matrix multiplication to be possible, the number of columns in Matrix A must match the number of rows in Matrix B. Here, both conditions are satisfied since Matrix A has 3 columns and Matrix B has 3 rows.
02

Set Up Resultant Matrix

The resultant matrix, denoted as Matrix C, will have the number of rows from Matrix A and the number of columns from Matrix B, resulting in a 2x2 matrix. We'll denote this matrix as \( C = \begin{bmatrix} c_{11} & c_{12} \ c_{21} & c_{22} \end{bmatrix} \).
03

Calculate Element \( c_{11} \)

The element \( c_{11} \) is calculated by taking the dot product of the first row of Matrix A \([2, 2, -1]\) and the first column of Matrix B \([0, -1, 0]\). This gives us: \( c_{11} = 2 \cdot 0 + 2 \cdot (-1) + (-1) \cdot 0 = 0 - 2 + 0 = -2 \).
04

Calculate Element \( c_{12} \)

The element \( c_{12} \) is calculated by taking the dot product of the first row of Matrix A \([2, 2, -1]\) and the second column of Matrix B \([2, 4, 2]\). This gives us: \( c_{12} = 2 \cdot 2 + 2 \cdot 4 + (-1) \cdot 2 = 4 + 8 - 2 = 10 \).
05

Calculate Element \( c_{21} \)

The element \( c_{21} \) is calculated by taking the dot product of the second row of Matrix A \([3, 0, 1]\) and the first column of Matrix B \([0, -1, 0]\). This gives us: \( c_{21} = 3 \cdot 0 + 0 \cdot (-1) + 1 \cdot 0 = 0 + 0 + 0 = 0 \).
06

Calculate Element \( c_{22} \)

The element \( c_{22} \) is calculated by taking the dot product of the second row of Matrix A \([3, 0, 1]\) and the second column of Matrix B \([2, 4, 2]\). This gives us: \( c_{22} = 3 \cdot 2 + 0 \cdot 4 + 1 \cdot 2 = 6 + 0 + 2 = 8 \).
07

Construct Resultant Matrix

Now that we have all the elements, the resulting matrix C is: \[ \begin{bmatrix} -2 & 10 \ 0 & 8 \end{bmatrix} \].

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Key Concepts

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

Matrix Dimensions
When it comes to matrix multiplication, the very first thing to check is the dimensions of the involved matrices. Knowing how to correctly identify and utilize matrix dimensions is crucial.
  • Matrix A has dimensions 2x3, meaning it has 2 rows and 3 columns.
  • Matrix B has dimensions 3x2, which indicates it has 3 rows and 2 columns.
In order for the multiplication of two matrices to be possible, the number of columns in the first matrix must be equal to the number of rows in the second matrix.
In this case, Matrix A has 3 columns while Matrix B has 3 rows. This means that the matrices are compatible for multiplication. If these conditions are not met, we would not be able to perform matrix multiplication.
Dot Product
To find each element of the resultant matrix, we need to perform a calculation called the dot product. The dot product is an essential concept in matrix multiplication.
It involves multiplying corresponding elements from a row of Matrix A and a column of Matrix B, then summing those products. This sum becomes the entry in the resultant matrix corresponding to that row and column.For example, to find the element at the first row and first column, or element \(c_{11}\), you take:
  • The first row of Matrix A: \([2, 2, -1]\)
  • The first column of Matrix B: \([0, -1, 0]\)
Then calculate the dot product as follows:
  • \(2 \cdot 0 + 2 \cdot (-1) + (-1) \cdot 0 = -2\)
This value, \(-2\), is the entry \(c_{11}\) in the resultant matrix. The same principle is applied to find every other entry in the resultant matrix.
Resultant Matrix
After calculating the dot products for each relevant row and column, you can compile these results into what is called the resultant matrix. The dimensions of this matrix are determined by the number of rows of the first matrix (Matrix A) and the number of columns of the second matrix (Matrix B).
In our case, Matrix A is 2x3, and Matrix B is 3x2. This results in a 2x2 resultant matrix because Matrix A has 2 rows and Matrix B has 2 columns.The final resultant matrix is:\[\begin{bmatrix} -2 & 10 \ 0 & 8 \end{bmatrix}\]Each element was calculated using the dot products as described earlier:
  • \( c_{11} = -2 \)
  • \( c_{12} = 10 \)
  • \( c_{21} = 0 \)
  • \( c_{22} = 8 \)
Understanding how to construct the resultant matrix is crucial as it鈥檚 the final outcome of your matrix multiplication process.

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Most popular questions from this chapter

A manufacturer of refrigerators must ship at least 100 refrigerators to its two West Coast warehouses. Each warehouse holds a maximum of 100 refrigerators. Warehouse A holds 25 refrigerators already, while warehouse \(B\) has 20 on hand. It costs \(\$ 12\) to ship a refrigerator to warehouse \(\mathrm{A}\) and \(\$ 10\) to ship one to warehouse B. How many refrigerators should be shipped to each warehouse to minimize cost? What is the minimum cost?

The break-even point for a company is the point where costs equal revenues. If both cost and revenue are expressed as linear equations, the break-even point is the solution of a linear system. In each exercise, \(C\) represents cost in dollars to produce x items, and R represents revenue in dollars from the sale of \(x\) items. Use the substitution method to find the break-even point in each case-that is, the point where \(C=R .\) Then find the value of \(C\) and \(R\) at that point. $$\begin{aligned}&C=20 x+10,000\\\&R=30 x-11,000\end{aligned}$$

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