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Chapter 4: Problem 3

Perform the indicated matrix operations. If the matrix does not exist, write impossible. $$ \left[\begin{array}{rr}{3} & {7} \\ {-2} & {1}\end{array}\right]-\left[\begin{array}{ll}{2} & {-3} \\ {5} & {-4}\end{array}\right] $$

Short Answer

Expert verified
\( \left(\begin{array}{cc} 1 & 10 \\ -7 & 5 \end{array}\right) \)

Step by step solution

01

Identify Matrix Dimensions

First, we need to check the dimensions of both matrices involved in the operation. Each matrix is a 2x2 matrix.
02

Check If Subtraction is Possible

Matrix subtraction requires the matrices to have the same dimensions. Since both matrices are 2x2, subtraction is possible.
03

Subtract the Matrices

Perform element-wise subtraction for each element in the corresponding positions of the two matrices. For the first element in the result matrix, subtract 2 from 3. Repeat for each corresponding element.
04

Calculate Each Element

Calculate each element in the resulting matrix: - Top left: 3 - 2 = 1 - Top right: 7 - (-3) = 10 - Bottom left: -2 - 5 = -7 - Bottom right: 1 - (-4) = 5
05

Write the Resulting Matrix

The resulting matrix, after performing the subtraction, is: \[\left(\begin{array}{cc}1 & 10 \-7 & 5\end{array}\right)\]

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Key Concepts

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

Understanding Matrix Dimensions
Matrix dimensions are critical when performing matrix operations. They determine whether you can add, subtract, or multiply matrices. Each matrix is defined by its number of rows and columns, presented as row x column. For example, a 2x2 matrix has 2 rows and 2 columns. In matrix subtraction, the matrices must have identical dimensions to be compatible for subtraction. If not, the operation is impossible.
When checking dimensions, look at the overall size of the matrix. This ensures each element can be matched to one in an identical position in the other matrix.
Understanding these dimensions is the first step in deciding what operations can be performed.
Element-Wise Operations in Matrix Subtraction
Element-wise operations refer to handling each corresponding matrix element separately, rather than considering the matrices as a whole. For subtraction, this means subtracting an element from the first matrix with its corresponding element from the second matrix.
This operation doesn't involve altering the structure of the matrices but focuses on matching elements through their positions. Each subtraction is performed using the formatting: \( A_{ij} - B_{ij} \), where \( A_{ij} \) and \( B_{ij} \) are elements at the i-th row and j-th column of matrices A and B.
Such element-wise handling makes the process straightforward and systematic, ensuring accuracy in operations.
Working with a 2x2 Matrix
A 2x2 matrix is one of the simplest forms of matrices and is structured with four elements. It is in the format where it has 2 rows and 2 columns. Each position in the matrix can be identified separately; for example:
  • First row, first column is the element at the top-left.
  • First row, second column is at the top-right.
  • Second row, first column is the bottom-left element.
  • Second row, second column is the bottom-right element.
This simplicity makes it easy to understand and manipulate when performing operations like addition, subtraction, or multiplication. Even though it looks compact, it illustrates the basic principles of larger and more complex matrix operations.
Executing Matrix Operations Effectively
Matrix operations involve performing calculations on matrices such as addition, subtraction, multiplication, and sometimes division. Each operation has its own set rules and conditions. Here, we focus on matrix subtraction, which requires matrices to be of the same size.
To effectively perform matrix subtraction, follow these steps:
  • Ensure both matrices involved have the same dimensions.
  • Perform element-wise operations, as discussed, matching each corresponding element and calculating their differences.
  • Check your results by verifying each subtraction to ensure accuracy.
These consistent rules across matrix subtraction make it a reliable mathematical tool used in various applications, from solving systems of equations to transformations in computer graphics.

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Most popular questions from this chapter

Solve each system of equations by graphing. \(y=3 x+5\) \(y=-2 x-5\)

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