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Chapter 6: Problem 66

Find each matrix product if possible. $$\left[\begin{array}{lll} 0 & 3 & -4 \end{array}\right]\left[\begin{array}{rrr} -2 & 6 & 3 \\ 0 & 4 & 2 \\ -1 & 1 & 4 \end{array}\right]$$

Short Answer

Expert verified
The product of the matrices is \([4, 8, -10]\).

Step by step solution

01

Confirm Matrix Dimensions Compatibility

The first matrix is a 1x3 matrix, and the second matrix is a 3x3 matrix. To multiply matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. Here, both numbers are 3, so the matrix multiplication is possible.
02

Set Up the Resulting Matrix

Since the first matrix is 1x3 and the second matrix is 3x3, the resulting matrix will be 1x3. It will contain 3 elements.
03

Compute Element in First Column

Multiply row elements of the first matrix with the corresponding column elements of the second matrix and sum them up: \[(0)(-2) + (3)(0) + (-4)(-1) = 0 + 0 + 4 = 4\]
04

Compute Element in Second Column

Multiply row elements of the first matrix with the corresponding column elements of the second matrix and sum them up:\[(0)(6) + (3)(4) + (-4)(1) = 0 + 12 - 4 = 8\]
05

Compute Element in Third Column

Multiply row elements of the first matrix with the corresponding column elements of the second matrix and sum them up: \[(0)(3) + (3)(2) + (-4)(4) = 0 + 6 - 16 = -10\]
06

Construct Resulting Matrix

The resulting matrix from our calculations is: \[\left[\begin{array}{lll} 4 & 8 & -10 \end{array}\right]\]

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Key Concepts

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

Matrix Dimensions
When discussing matrix multiplication, the first thing to understand is the dimensions of the matrices involved. A matrix is essentially a grid of numbers organized in rows and columns. Each matrix has dimensions denoted as "rows 脳 columns". For successful multiplication, the number of columns in the first matrix must match the number of rows in the second matrix.
  • In the exercise example, the first matrix is a 1x3 matrix, meaning it has 1 row and 3 columns.
  • The second matrix is a 3x3 matrix, meaning it has 3 rows and 3 columns.
Since the first matrix has 3 columns and the second matrix has 3 rows, matrix multiplication is possible according to the rule.
Resulting Matrix
Understanding the size of the resulting matrix is key in the multiplication process. When two matrices are multiplied, the dimensions of the resulting matrix depend on the rows of the first matrix and the columns of the second matrix.
  • The resulting matrix will have dimensions equal to the number of rows of the first matrix by the number of columns of the second matrix.
  • In this example, the first matrix has 1 row and the second matrix has 3 columns.
  • Thus, the resulting matrix will be a 1x3 matrix, meaning it will have one row with three elements.
This tells us what shape the solution of our matrix multiplication will take, crucial for organizing our results appropriately.
Element Computation
The computation of each element in the resulting matrix involves using the elements of the original matrices. Specifically, each element in the resulting matrix is obtained by the dot product of the corresponding row of the first matrix and the column of the second matrix.

Let's break it down from the example:
  • First Element: We multiply and sum: \( (0)(-2) + (3)(0) + (-4)(-1) = 4 \)
  • Second Element: We perform the same process: \( (0)(6) + (3)(4) + (-4)(1) = 8 \)
  • Third Element: Similarly, we compute: \( (0)(3) + (3)(2) + (-4)(4) = -10 \)
Each element involves multiplying the corresponding components and summing them up, as seen above.
This ensures every computation in the resulting matrix is accurately calculated, leading to the final matrix: \[\left[\begin{array}{lll} 4 & 8 & -10 \end{array}\right]\]

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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?

Graph the solution set of each system of inequalities by hand. $$\begin{array}{r} x+y \leq 36 \\ -4 \leq x \leq 4 \end{array}$$

The table shows weight \(W,\) neck size \(N,\) overall length \(L,\) and chest size \(C\) for four bears. $$\begin{array}{|c|c|c|c|} \hline W \text { (pounds) } & N \text { (inches) } & L \text { (inches) } & C \text { (inches) } \\ \hline 125 & 19 & 57.5 & 32 \\\ 316 & 26 & 65 & 42 \\ 436 & 30 & 72 & 48 \\ 514 & 30.5 & 75 & 54 \end{array}$$ A. We can model these data with the equation $$ W=a+b N+c L+d C $$ where \(a, b, c,\) and \(d\) are constants. To do so, represent a system of linear equations by a \(4 \times 5\) augmented matrix whose solution gives values for \(a, b, c,\) and \(d\) B. Solve the system. Round each value to the nearest thousandth. C. Predict the weight of a bear with \(N=24, L=63\) and \(C=39 .\) Interpret the result.

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} &y \leq\left(\frac{1}{2}\right)^{x}\\\ &y \geq 4 \end{aligned}$$

Shade the region(s) contained inside the graphs and give any points of intersection of the equations. $$\begin{aligned} &y=2 x-1\\\ &y=2-x^{2} \end{aligned}$$
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