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Chapter 8: Problem 1

fill in the blanks. Two matrices are ________ when their corresponding entries are equal.

Short Answer

Expert verified
Equal

Step by step solution

01

Understanding Matrices and their properties

Matrices are defined as a rectangular arrangement of numbers in rows and columns. They are considered equal when each element in the corresponding position in both matrices is equal.
02

Filling the information into the gap

The term that describes the condition where corresponding entries of two matrices are equal is 'equal'. Thus, the completed sentence reads as: Two matrices are 'equal' when their corresponding entries are equal.

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Key Concepts

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

Matrix Properties
When studying matrices, understanding their properties is key to grasping more complex concepts. One fundamental property is matrix equality. This property states that two matrices are considered equal if and only if they have the same dimensions and each corresponding element within the matrices is identical.

In practice, checking matrix equality involves comparing each component:
  • **Same Dimensions**: Both matrices must have the same number of rows and columns.
  • **Identical Elements**: Each element from matrix A must match the corresponding element in matrix B (e.g., the element in the first row and first column of A must be equal to the element in the first row and first column of B).
These properties are foundational in matrix algebra and prove important for solving many linear algebra problems.
Matrices
Matrices play a critical role in mathematics and various applications such as computer graphics, statistics, and engineering. At its core, a matrix is an ordered array of numbers laid out in rows and columns. This arrangement enables us to perform various calculations and transformations efficiently.

Matrices are often represented using capital letters (such as A, B, or C) and are defined by their size or order, which is noted by their number of rows (m) and columns (n). Therefore, a matrix A with 3 rows and 4 columns is an example of a 3x4 matrix.
  • **Row Matrix**: A matrix with only one row.
  • **Column Matrix**: A matrix with only one column.
  • **Square Matrix**: A matrix with the same number of rows and columns.
Each type of matrix has unique properties and uses, making them indispensable tools across different fields.
Rectangular Arrangement of Numbers
A matrix is often referred to as a rectangular arrangement of numbers, emphasizing its grid-like structure. This format consists of horizontal rows and vertical columns that organize data systematically.

The rectangular arrangement simplifies various types of mathematical computations, such as:
  • **Addition**: Matrices can be added together if they possess identical dimensions, leading to a new matrix whose elements are the sum of corresponding elements.
  • **Multiplication**: Matrix multiplication involves multiplying rows with columns, resulting in a new matrix that's potentially of a different size according to specific rules.
These operations highlight the versatility and efficiency of using matrices as a mathematical and practical tool for handling complex sets of data.

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Most popular questions from this chapter

Solve for \(x\). $$\left|\begin{array}{rr} x & 4 \\ -1 & x \end{array}\right|=20$$

Solve for \(x\). $$\left|\begin{array}{ll} x & 2 \\ 1 & x \end{array}\right|=2$$

use the matrices $$A=\left[\begin{array}{rr} 2 & -1 \\ 1 & 3 \end{array}\right] \quad \text { and } \quad B=\left[\begin{array}{rr} -1 & 1 \\ 0 & -2 \end{array}\right].$$ $$\text { Show that }(A+B)^{2} \neq A^{2}+2 A B+B^{2}$$.

Consider the circuit shown in the figure. The currents \(I_{1}, I_{2},\) and \(I_{3}\), in amperes, are the solution of the system of linear equations. $$\left\\{\begin{aligned}2 I_{1}\quad\quad\quad &+4 I_{3}=E_{1} \\\I_{2}+4 I_{3} &=E_{2} \\\I_{1}+I_{2}-I_{3} &=0\end{aligned}\right.$$ where \(E_{1}\) and \(E_{2}\) are voltages. Use the inverse of the coefficient matrix of this system to find the unknown currents for the given voltages. $$\begin{aligned}&E_{1}=28 \text { volts }\\\&E_{2}=21 \text { volts }\end{aligned}$$

Solve for \(x\). $$\left|\begin{array}{cc} x+1 & 2 \\ -1 & x \end{array}\right|=4$$
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