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Chapter 16: Problem 11

Find the indicated sums of matrices. $$\left[\begin{array}{rr} 2 & 3 \\ -5 & 4 \end{array}\right]+\left[\begin{array}{rr} -1 & 7 \\ 5 & -2 \end{array}\right]$$

Short Answer

Expert verified
The sum of the two matrices is \(\begin{bmatrix} 1 & 10 \\ 0 & 2 \end{bmatrix}\).

Step by step solution

01

Understand the Problem

We need to add two matrices together. The dimensions of the matrices are both 2x2, which allows us to add them.
02

Add Corresponding Elements

For matrix addition, we add the elements at corresponding positions in each matrix. Let鈥檚 do this one by one.
03

Add Elements in the First Row, First Column

Add the elements from the first row, first column: \(2 + (-1) = 1\).
04

Add Elements in the First Row, Second Column

Add the elements from the first row, second column: \(3 + 7 = 10\).
05

Add Elements in the Second Row, First Column

Add the elements from the second row, first column: \(-5 + 5 = 0\).
06

Add Elements in the Second Row, Second Column

Add the elements from the second row, second column: \(4 + (-2) = 2\).
07

Combine Results into a New Matrix

The resulting matrix after adding corresponding elements is: \[ \left[ \begin{array}{rr} 1 & 10 \ 0 & 2 \end{array} \right] \]

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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 a simple and commonly used type of matrix in mathematics. It is called a 2x2 matrix because it contains two rows and two columns, making it a square matrix. Here's how you can identify a 2x2 matrix:
  • It has 2 rows.
  • It has 2 columns.
  • Every element within the matrix is arranged as if on a grid or in a table.
Each number in the matrix is called an "element". When we write a 2x2 matrix, it looks like this: \[ \begin{array}{rr} a & b \ c & d \end{array} \] In this representation, \(a, b, c,\) and \(d\) are the elements of the matrix. Understanding 2x2 matrices is fundamental before you move on to more complex types of matrices.
element-wise addition
Element-wise addition is a basic operation that is fundamental to understanding how to add matrices. When performing matrix addition, you add each element from one matrix to the corresponding element in another matrix. Let's break it down:
  • Each element in the first matrix is paired with an element in the same position in the second matrix.
  • Only matrices of the same size can be added together, meaning both must be 2x2, 3x3, etc.
  • The result of adding two matrices of the same size is another matrix of the same size.
For example, if you have two matrices:\[ \begin{array}{rr} 2 & 3 \ -5 & 4 \end{array} \] and \[ \begin{array}{rr} -1 & 7 \ 5 & -2 \end{array} \],You perform element-wise addition to get:\(2 + (-1), 3 + 7, -5 + 5,\) and \(4 + (-2)\),which results in the new matrix:\[ \begin{array}{rr} 1 & 10 \ 0 & 2 \end{array} \] Take care to always pair elements correctly by their position in the matrix.
matrix operations
Matrix operations are a wide and exciting area of mathematics that allow us to perform various calculations across rows and columns of matrices, affecting all elements they contain. Matrix addition is one of the simplest operations, but there are more:
  • **Matrix Subtraction**: Similar to addition but subtracts corresponding elements.
  • **Matrix Multiplication**: More complex and requires multiplying rows by columns, rather than element-wise.
  • **Scalar Multiplication**: Multiplying every element by a constant number (called a scalar).
  • **Transpose**: Flipping a matrix over its diagonal, turning rows into columns and vice versa.
When performing operations like matrix addition, it is crucial to ensure all matrices involved have matching dimensions. Failing to do so will result in undefined operations. Grasping matrix operations is essential for further studies in linear algebra, as these concepts are building blocks for solving real-world problems involving complex data structures.

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Most popular questions from this chapter

Solve the given systems of equations by using the inverse of the coefficient matrix. Use a calculator to perform the necessary matrix operations and display the results and the check. See Example 5. $$\begin{aligned}&2 v+3 w+x-y-2 z=6\\\&6 v-2 w-x+3 y-z=21\\\&v+3 w-4 x+2 y+3 z=-9\\\&3 v-w-x+7 y+4 z=5\\\&v+6 w+6 x-4 y-z=-4\end{aligned}$$

Solve the given problems by using determinants. Three computer programs, \(A, B\), and \(C\), use \(15 \%\) of a computer's \(6.0 \mathrm{GB}\) (gigabyte) hard-drive memory. If two additional programs are added, one requiring the same memory as \(A\) and the other half the memory of \(C, 22 \%\) of the memory will be used. However, if two other programs are added to \(A, B,\) and \(C,\) one requiring half the memory of \(B\) and the other the same memory as \(C, 25 \%\) of the memory will be used. How many megabytes of memory are required for each of \(A, B,\) and \(C ?\)

Perform the indicated matrix multiplications on a calculator, using the following matrices. For matrix \(A, A^{2}=A \times A.\) $$A=\left[\begin{array}{rrr}2 & -3 & -5 \\\\-1 & 4 & 5 \\\1 & -3 & -4\end{array}\right] B=\left[\begin{array}{rrr}1 & -2 & -6 \\\\-3 & 2 & 9 \\\2 & 0 & -3\end{array}\right] C=\left[\begin{array}{rrr}1 & -3 & -4 \\\\-1 & 3 & 4 \\\1 & -3 & -4\end{array}\right]$$ Show that \(C^{2}=O\)

Solve the indicated systems of equations using the inverse of the coefficient matrix. It is necessary to set up the appropriate equations. A river tour boat takes \(5.0 \mathrm{h}\) to cruise downstream and \(7.0 \mathrm{h}\) for the return upstream. If the river flows at \(4.0 \mathrm{mi} / \mathrm{h}\), how fast does the boat travel in still water, and how far downstream does the boat go before starting the return trip?

Solve the given problems by using determinants. An alloy is to be made from four other alloys containing copper (Cu), nickel (Ni), zinc (Zn), and Iron (Fe). The first is 80\% Cu and 20\% Ni. The second is 60\% Cu, 20\% Ni, and 20\% Zn. The third is \(30 \%\) Cu, \(60 \%\) Ni, and \(10 \%\) Fe. The fourth is \(20 \%\) Ni, \(40 \% \mathrm{Zn},\) and \(40 \% \mathrm{Fe} .\) How much of each is needed so that the final alloy has \(56 \mathrm{g} \mathrm{Cu}, 28 \mathrm{g} \mathrm{Ni}, 10 \mathrm{g} \mathrm{Zn},\) and \(6 \mathrm{g} \mathrm{Fe} ?\)
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