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Chapter 6: Problem 2

Let \(A=\left[\begin{array}{ll}4 & -1 \\ 7 & -9\end{array}\right], B=\left[\begin{array}{rr}3 & 9 \\ 2 & -2\end{array}\right], C=\left[\begin{array}{rr}8 & 3 \\ 0 & -3\end{array}\right]\) \(D=\left[\begin{array}{rrr}5 & -6 & 1 \\ 10 & 3 & -1\end{array}\right], E=\left[\begin{array}{rr}-7 & 4 \\ -3 & 2 \\ 2 & -1\end{array}\right]\) \(F=\left[\begin{array}{rrr}-4 & 2 & 3 \\ 0 & -1 & 2 \\ -7 & -2 & 5\end{array}\right]\), and \(v=\left[\begin{array}{r}2 \\\ -3\end{array}\right]\). Compute \(A+C\).

Short Answer

Expert verified
\(A + C = \begin{bmatrix} 12 & 2 \ 7 & -12 \end{bmatrix}\)

Step by step solution

01

Identify the matrices

We are given the matrices \(A\) and \(C\). Matrix \(A\) is: \[A = \begin{bmatrix} 4 & -1 \ 7 & -9 \end{bmatrix}\] and matrix \(C\) is: \[C = \begin{bmatrix} 8 & 3 \ 0 & -3 \end{bmatrix}\]
02

Verify dimensions

Both matrices \(A\) and \(C\) are of size 2x2. Since they have the same dimensions, we can add them element-wise.
03

Add corresponding elements

To find the sum \(A + C\), add the corresponding elements from matrices \(A\) and \(C\): \(A + C = \begin{bmatrix} 4 & -1 \ 7 & -9 \end{bmatrix} + \begin{bmatrix} 8 & 3 \ 0 & -3 \end{bmatrix}\)
04

Perform the addition

Calculate the sum of each corresponding element:\[A + C = \begin{bmatrix} 4+8 & -1+3 \ 7+0 & -9+(-3) \end{bmatrix} = \begin{bmatrix} 12 & 2 \ 7 & -12 \end{bmatrix}\]

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Key Concepts

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

element-wise addition
When we perform matrix addition, we are doing it in an element-wise manner. This means we directly add each corresponding element from one matrix to the other.
For the matrices A and C, their corresponding elements are added as shown in the following steps:
- The element in the first row, first column of A (which is 4) is added to the element in the first row, first column of C (which is 8). This results in 12.
- Similarly, the element in the first row, second column of A (which is -1) is added to the element in the first row, second column of C (which is 3). This gives us 2.
You would do the same for all the positions of the matrices, ensuring each element from A is added to the matched element from C.
It is important to note that matrix addition follows this simple principle of corresponding elements sum. Any attempt to add matrices of different dimensions would be incorrect, as there would be no matching elements to pair and sum.
matrix dimensions
In linear algebra, the dimensions of a matrix鈥攊ndicated as rows x columns鈥攑lay a critical role. Dimensions determine whether operations like addition and multiplication are valid.
In the given exercise, both matrices A and C are of dimension 2x2. This makes it possible for them to be added together. Think of the dimensions as the 'shape' of the matrix; for example, a 2x2 matrix has 2 rows and 2 columns, forming a square matrix.
It's crucial to always verify the dimensions of any matrices involved in operations. If the dimensions are not the same, addition can't be performed. This verification step ensures the matrices conform to the rules of linear algebra.
linear algebra
Linear algebra is a branch of mathematics that studies vectors, matrices, and linear transformations. It is foundational for multiple fields including physics, engineering, computer science, and statistics.
Matrices are key components in linear algebra. They can represent systems of linear equations, transform geometric data, and much more. Operations like matrix addition, multiplication, and inversion are fundamental.
The given exercise is a simple example of how matrices are used and manipulated in linear algebra. It highlights how the dimensions and corresponding elements govern the feasibility and outcome of matrix operations.
2x2 matrices
A 2x2 matrix is a matrix with exactly 2 rows and 2 columns. This small-sized matrix is often used in teaching and solving basic linear algebra problems.
The structure is straightforward and allows for easy computation of operations such as addition, subtraction, and multiplication.
In the example given, matrices A and C are both 2x2 matrices, making them perfect candidates for demonstrating element-wise addition. The resultant matrix also retains the dimensions 2x2, ensuring the operation adheres to linear algebra norms.
Understanding 2x2 matrices is an excellent stepping stone for diving into more complex matrix operations and higher-dimensional matrices.

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Most popular questions from this chapter

San Diego fairy shrimp live in ponds that fill and dry many times during a year. While the ponds are dry, the [airy shrimp survive as cysts. The population is divided into three groups. Cysts that survive one dry period are in group 1 cysts that survive two dry periods are in group 2 and cysts that survive three or more dry periods are in group \(3 .\) The Leslie matrix representing the survivability and fecundity is given below. \({ }^{14}\) $$G=\left[\begin{array}{ccc} 3.6 & 0.98 & 0.65 \\ 0.5 & 0 & 0 \\ 0 & 0.5 & 0.49 \end{array}\right]$$ In the third dry period there are 8365,1095, and 310 individuals in groups 1,2, and 3, respectively. a) Find the inverse of the Leslie matrix. b) Estimate the population of each group during the second dry spell. c) Estimate the population of each group during the first dry spell.

Write the vector \(v\) as a linear combination of the vectors \(\mathbf{w}\) and \(\mathbf{u}\). $$ \mathbf{v}=\left[\begin{array}{l} 5 \\ 5 \end{array}\right], \mathbf{w}=\left[\begin{array}{l} 4 \\ 1 \end{array}\right], \mathbf{u}=\left[\begin{array}{l} 3 \\ 2 \end{array}\right] $$

In the absence of competitors and herbivores, plant growth can be modeled by the recursion relation $$ M_{n+1}=\frac{(1+\rho) M_{n}}{1+\theta M_{n}} $$ where \(M_{n}\) is the total plant mass after \(3 n\) days, \(\rho\) is the maximum growth rate, and \(\theta\) is a constant. For a plant, \(\rho=0.3, \theta=0.001\), and the starting mass is \(M_{0}=1 \mathrm{~g}\) a) Plot the points \(\left(n, x_{n}\right)\) for \(n=0,1,2,3, \ldots, 40\). b) Does this graph look familiar?

The matrix \(A\) has eigenvalues \(r_{1}\) and \(r_{2}\) with corresponding eigenvectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\), respectively. Compute \(\mathbf{A}^{n} \mathbf{w}\). $$ \begin{array}{l} r_{1}=3, r_{2}=0, v_{1}=\left[\begin{array}{r} 1 \\ -1 \end{array}\right], v_{2}=\left[\begin{array}{l} 1 \\ 1 \end{array}\right], n=4, \\ w=\left[\begin{array}{l} 2 \\ 4 \end{array}\right] \end{array} $$

Dinophilus gyrociliatus is a small species that lives in the fouling community of harbor environments. On average, a female has approximately 30 eggs during her first 6 wk of life. If she survives her first \(6 \mathrm{wk}\), she has on average 15 eggs her second 6 wk of life. Furthermore, approximately \(80 \%\) of the females survive their first \(6 \mathrm{wk}\) and none survive beyond the second \(6 \mathrm{wk} .{ }^{+}\) Assume half the eggs are female and for simplicity, assume that all the eggs are hatched at once at the beginning of each 6-wk period. Ignore the male population and make the two groups females under 6 wk old and females over 6 wk old. a) Draw and label the Leslie diagram. b) Find the Leslie matrix. c) Twenty hatchlings are introduced into an area. Estimate the population of the two groups after 6 wk. d) Estimate the population of the two groups after 12 wk.
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