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Chapter 9: Problem 53

Find each matrix product when possible. $$\left[\begin{array}{rrr} 3 & -4 & 1 \\ 5 & 0 & 2 \end{array}\right]\left[\begin{array}{r} -1 \\ 4 \\ 2 \end{array}\right]$$

Short Answer

Expert verified
The product is \[ \begin{bmatrix} -17 \ -1 \end{bmatrix} \].

Step by step solution

01

- Identify the dimensions

Confirm the dimensions of the matrices. The first matrix is a 2x3 matrix (2 rows and 3 columns), while the second matrix is a 3x1 matrix (3 rows and 1 column).
02

- Check for compatibility

Verify if the matrices can be multiplied. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. In this case, both matrices are compatible because the first matrix has 3 columns and the second matrix has 3 rows.
03

- Perform matrix multiplication

Multiply each element of the rows of the first matrix by the corresponding element of the columns of the second matrix, then sum the products.Let's denote the first matrix as \[A = \begin{bmatrix} 3 & -4 & 1 \ 5 & 0 & 2 \end{bmatrix}\] and the second matrix as \[B = \begin{bmatrix} -1 \ 4 \ 2 \end{bmatrix}\]To find the product matrix, calculate each element as follows:\[ C_{11} = 3(-1) + (-4)(4) + 1(2) = -3 - 16 + 2 = -17\]\[ C_{21} = 5(-1) + 0(4) + 2(2) = -5 + 0 + 4 = -1\]So the resulting matrix C is:\[ C = \begin{bmatrix} -17 \ -1 \end{bmatrix} \]

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Key Concepts

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

matrix dimensions
Matrix dimensions are an essential concept in matrix multiplication. They indicate the number of rows and columns a matrix has. For example, in the original exercise, we have two matrices. The first matrix has dimensions 2x3, meaning it has 2 rows and 3 columns. The second matrix is a 3x1 matrix, having 3 rows and 1 column.
Understanding matrix dimensions helps you visualize the structure of a matrix. It's like knowing the length and width of a rectangle before drawing it.
When writing matrix dimensions, always write the number of rows first, followed by the number of columns. So, a matrix with 4 rows and 5 columns is written as 4x5.
Getting the dimensions correct is the first step to performing any matrix operations correctly.
compatibility check
Before performing matrix multiplication, it's crucial to check if the matrices are compatible. This step ensures that matrix multiplication is possible between the two matrices.
For two matrices to be compatible for multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix. In our exercise, the first matrix has 3 columns, and the second matrix has 3 rows. Therefore, they are compatible.
To perform this check, simply compare the second dimension of the first matrix with the first dimension of the second matrix. If they match, you can proceed with multiplication.
If the matrices are not compatible, you cannot multiply them, and attempting to do so would lead to errors or undefined results.
element-wise product
The actual multiplication in matrix multiplication involves taking the element-wise product. This means you multiply corresponding elements from the rows of the first matrix by elements from the columns of the second matrix.
For the matrices in our exercise, take each element from a row in the first matrix and multiply it by the corresponding element in the column of the second matrix. Then, sum these products to get a single entry in the resulting matrix.
Here's how it works for our example:
  • For the first entry: \( C_{11} = 3(-1) + (-4)(4) + 1(2) = -3 - 16 + 2 = -17 \)
  • For the second entry: \( C_{21} = 5(-1) + 0(4) + 2(2) = -5 + 0 + 4 = -1 \)
This method ensures that each entry in the result matrix is calculated by considering all elements from the related row and column, leading to the final product matrix.
That's it! Remember these steps, and you'll master the art of matrix multiplication in no time.

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Solve each problem. Several years ago, mathematical ecologists created a model to analyze population dynamics of the endangered northern spotted owl in the Pacific Northwest. The ecologists divided the female owl population into three categories: juvenile (up to \(1 \text { yr old }),\) subadult \((1\) to 2 yr old ) and adult (over 2 yr old). They concluded that the change in the makeup of the northern spotted owl population in successive years could be described by the following matrix equation. $$ \left[\begin{array}{c} j_{n+1} \\ s_{n+1} \\ a_{n+1} \end{array}\right]=\left[\begin{array}{rrr} 0 & 0 & 0.33 \\ 0.18 & 0 & 0 \\ 0 & 0.71 & 0.94 \end{array}\right]\left[\begin{array}{c} j_{n} \\ s_{n} \\ a_{n} \end{array}\right] $$ The numbers in the column matrices give the numbers of females in the three age groups after \(n\) years and \(n+1\) years. Multiplying the matrices yields the following. \(j_{n+1}=0.33 a_{n}\) Each year 33 juvenile females are born for each 100 adult females. \(s_{n+1}=0.18 j_{n}\) Each year 18\% of the juvenile females survive to become subadults. \(a_{n+1}=0.71 s_{n}+0.94 a_{n} \quad\) Each year \(71 \%\) of the subadults survive to become adults, and \(94 \%\) of the adults survive. (a) Suppose there are currently 3000 female northern spotted owls made up of 690 juveniles, 210 subadults, and 2100 adults. Use the matrix equation on the preceding page to determine the total number of female owls for each of the next 5 yr. (b) Using advanced techniques from linear algebra, we can show that in the long run, $$ \left[\begin{array}{c} j_{n+1} \\ s_{n+1} \\ a_{n+1} \end{array}\right] \approx 0.98359\left[\begin{array}{c} j_{n} \\ s_{n} \\ a_{n} \end{array}\right] $$ What can we conclude about the long-term fate of the northern spotted owl? (c) In the model, the main impediment to the survival of the northern spotted owl is the number 0.18 in the second row of the 3 \(\times 3\) matrix. This number is low for two reasons. The first year of life is precarious for most animals living in the wild. In addition, juvenile owls must eventually leave the nest and establish their own territory. If much of the forest near their original home has been cleared, then they are vulnerable to predators while searching for a new home. Suppose that, thanks to better forest management, the number 0.18 can be increased to \(0.3 .\) Rework part (a) under this new assumption.
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