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Chapter 9: Problem 3

Perform the matrix operation, or if it is impossible, explain why. $$\left[\begin{array}{rr} 2 & 6 \\ -5 & 3 \end{array}\right]+\left[\begin{array}{rr} -1 & -3 \\ 6 & 2 \end{array}\right]$$

Short Answer

Expert verified
The resulting matrix is \(\begin{bmatrix} 1 & 3 \\ 1 & 5 \end{bmatrix}\).

Step by step solution

01

Identify the Size of the Matrices

Both matrices have the size of 2 rows and 2 columns, meaning they are 2x2 matrices. Matrix addition requires both matrices to have the same dimensions for the operation to be possible.
02

Add the Corresponding Elements

Since the matrices have the same dimensions, we add their corresponding elements together. The resulting matrix will also be a 2x2 matrix.\[\begin{bmatrix}2 + (-1) & 6 + (-3) \-5 + 6 & 3 + 2\end{bmatrix}= \begin{bmatrix}1 & 3 \1 & 5\end{bmatrix}\]
03

Write Down the Resulting Matrix

After adding the corresponding elements, the resulting matrix is \[\begin{bmatrix}1 & 3 \1 & 5\end{bmatrix}\].

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Key Concepts

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

2x2 Matrices
A 2x2 matrix is one of the simplest forms of a matrix in linear algebra. It consists of two rows and two columns, and can be represented as follows:
  • The top row is composed of two elements, which we'll call \(a\) and \(b\).
  • The bottom row also contains two elements, \(c\) and \(d\).
This matrix can be written in the standard form: \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]This type of matrix allows for straightforward calculations and is often the starting point for learning more complex matrix operations. Each matrix element is typically an integer or a real number.
Matrix Operations
Matrix operations include a variety of manipulations such as addition, subtraction, and multiplication. Matrix addition is one of the simplest operations, requiring only that two matrices share the same dimensions.

To add two matrices, you simply add each corresponding element from the first matrix to the element in the same position in the second matrix. Applying this to 2x2 matrices:
  • Add element \(a\) from the first matrix to element \(e\) in the second matrix to get the new first-row, first-column element.
  • Continue this process for each corresponding pair of elements: \(b\) with \(f\), \(c\) with \(g\), and \(d\) with \(h\).
The resulting matrix from this process is another 2x2 matrix. This operation is associative and commutative, meaning you can add matrices in any order and still get the same result.
Matrix Dimensions
Matrix dimensions are crucial when it comes to performing operations. The dimensions of a matrix are indicated by the number of rows and columns it contains. For instance, a 2x2 matrix has 2 rows and 2 columns.

Understanding matrix dimensions is vital because:
  • They determine the possibility of performing operations like addition. Two matrices must have the same dimensions to be added together.
  • They influence the structure and shape of the matrix, impacting how you visually interpret and apply them in calculations.
  • They are foundational for understanding more complex topics like matrix multiplication or transformations.
Identifying and remembering these dimensions simplifies matrix operations and makes it easier to avoid errors in your calculations.

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Most popular questions from this chapter

Sales Commissions An encyclopedia saleswoman works for a company that offers three different grades of bindings for its encyclopedias: standard, deluxe, and leather. For each set she sells, she earns a commission based on the set's binding grade. One week she sells one standard, one deluxe, and two leather sets and makes \(\$ 675\) in commission. The next week she sells two standard, one deluxe, and one leather set for a \(\$ 600\) commission. The third week she sells one standard, two deluxe, and one leather set, carning \(\$ 625\) in commission.

(a) Suppose that \(\left(x_{0}, y_{0}, z_{0}\right)\) and \(\left(x_{1}, y_{1}, z_{1}\right)\) are solutions of the system $$ \left\\{\begin{array}{l} a_{1} x+b_{1} y+c_{1} z=d_{1} \\ a_{2} x+b_{2} y+c_{2} z=d_{2} \\ a_{3} x+b_{3} y+c_{3} z=d_{3} \end{array}\right. $$ Show that \(\left(\frac{x_{0}+x_{1}}{2}, \frac{y_{0}+y_{1}}{2}, \frac{z_{0}+z_{1}}{2}\right)\) is also a solution. (b) Use the result of part (a) to prove that if the system has two different solutions, then it has infinitely many solutions.

Find the partial fraction decomposition of the rational function. $$\frac{7 x-3}{x^{3}+2 x^{2}-3 x}$$

Evaluate the determinants. $$\left|\begin{array}{lllll} a & a & a & a & a \\ 0 & a & a & a & a \\ 0 & 0 & a & a & a \\ 0 & 0 & 0 & a & a \\ 0 & 0 & 0 & 0 & a \end{array}\right|$$

Find the inverse of the matrix. For what value(s) of \(x\) if any, does the matrix have no inverse? $$\left[\begin{array}{ll} \sec x & \tan x \\ \tan x & \sec x \end{array}\right]$$
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