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Chapter 3: Problem 58

What would a matrix \(A\) look like if \(A_{i j}=0\) whenever \(i \neq j\) ?

Short Answer

Expert verified
A matrix \(A\) with the condition \(A_{ij} = 0\) when \(i \neq j\) would be a diagonal matrix. It would have the general form: \(A = \begin{bmatrix} a_{11} & 0 & \dots & 0 \\ 0 & a_{22} & \dots & 0 \\\vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & a_{nn} \end{bmatrix}\)

Step by step solution

01

Understand the condition

The condition given is A_{ij} = 0 when i 鈮� j, which means that if the row number (i) is not equal to the column number (j), the element at that position in the matrix must be zero.
02

Picture the matrix

Now let's visualize what happens to the matrix A when all the off-diagonal elements are zero. Take an arbitrary 3x3 matrix A as an example: \(A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}\)
03

Apply the condition

Now apply the condition A_{ij} = 0 if i 鈮� j: A becomes: \(A = \begin{bmatrix} a_{11} & 0 & 0 \\ 0 & a_{22} & 0 \\ 0 & 0 & a_{33} \end{bmatrix}\)
04

Generalize for any matrix size

Now you can generalize this pattern for any square matrix of size nxn: \(A = \begin{bmatrix} a_{11} & 0 & \dots & 0 \\ 0 & a_{22} & \dots & 0 \\\vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & a_{nn} \end{bmatrix}\)
05

Identify the matrix type

The matrix A with A_{ij} = 0 for i 鈮� j is called a diagonal matrix.

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Key Concepts

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

Matrix Algebra
Matrix algebra is a branch of mathematics that deals with operations on matrices and their properties. It involves various operations such as addition, subtraction, multiplication, and finding inverses of matrices. Matrices are rectangular arrays of numbers, symbols, or expressions arranged in rows and columns.
The fundamental operations include:
  • Addition: Two matrices can be added if they have the same dimensions. The elements in corresponding positions are added together individually.
  • Multiplication: Matrix multiplication is more complex than addition or subtraction. The number of columns in the first matrix must match the number of rows in the second matrix. The element at row i and column j in the resulting matrix is the sum of the products of corresponding elements from the row of the first matrix and the column of the second matrix.
  • Scalar Multiplication: Every element of the matrix is multiplied by a scalar (a constant value).
  • Transpose: The transpose of a matrix is obtained by swapping its rows and columns.
Understanding matrix algebra is essential for working with more advanced matrix types, like a diagonal matrix, which simplifies various operations.
Square Matrix
A square matrix is a type of matrix where the number of rows is equal to the number of columns, represented as an n x n matrix. This symmetry makes square matrices a fundamental concept in linear algebra and matrix theory.
Some key characteristics of square matrices include:
  • Identity Matrix: It is a square matrix with all the diagonal elements equal to 1 and the rest of the elements are 0.
  • Determinant: Only square matrices have a well-defined determinant, a value that can be computed from its elements and reflects certain properties of the matrix.
  • Diagonal Matrix: A special kind of square matrix where all elements off the main diagonal are zero, making it a useful tool for simplifying calculations.
  • Invertible Matrix: A square matrix can be inverted only if its determinant is non-zero.
Square matrices play a crucial role in mathematical applications, such as solving systems of linear equations and exploring transformations in geometry.
Matrix Elements
Matrix elements are the individual items or numbers that make up a matrix. These elements are arranged in rows and columns, and each element is identified by its position within the matrix, denoted as \(A_{ij}\) where i is the row number and j is the column number.
Key points about matrix elements include:
  • Diagonal Elements: Elements where the row number equals the column number. In a diagonal matrix, only these elements are non-zero.
  • Off-Diagonal Elements: In a matrix, any element where the row number is not equal to the column number. These are zero in a diagonal matrix.
  • Zero Matrix: A matrix where all elements are zero is called a zero or null matrix.
  • Symmetric Matrix: A matrix is symmetric if it equals its transpose, implying \(A_{ij} = A_{ji}\).
Understanding the placement and significance of matrix elements is crucial for manipulating and interpreting matrices, especially when ensuring conditions like those of a diagonal matrix.

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Most popular questions from this chapter

I Population Movement In 2006, the population of the United States, broken down by regions, was \(55.1\) million in the Northeast, \(66.2\) million in the Midwest, \(110.0\) million in the South, and \(70.0\) million in the West. \({ }^{19}\) The matrix \(P\) below shows the population movement during the period 2006 \(2007 .\) (Thus, \(98.92 \%\) of the population in the Northeast stayed there, while \(0.17 \%\) of the population in the Northeast moved to the Midwest, and so on.) \(\begin{array}{ccll}\text { To } & \text { To } & \text { To } & \text { To } \\\ \text { NE } & \text { MW } & \text { S } & \text { W }\end{array}\) \(P=\begin{aligned}&\text { From NE } \\\&\text { From MW } \\\&\text { From S } \\\&\text { From W }\end{aligned}\left[\begin{array}{llll}0.9892 & 0.0017 & 0.0073 & 0.0018 \\ 0.0010 & 0.9920 & 0.0048 & 0.0022 \\ 0.0018 & 0.0024 & 0.9934 & 0.0024 \\ 0.0008 & 0.0033 & 0.0045 & 0.9914\end{array}\right]\) Set up the 2006 population figures as a row vector. Assuming that these percentages also describe the population movements from 2005 to 2006 , show how matrix inversion and multiplication allow you to compute the population in each region in 2005 . (Round all answers to the nearest \(0.1\) million.)

Can an external demand be met by an economy whose technology matrix \(A\) is the identity matrix? Explain.

Evaluate the given expression. Take$$\begin{aligned}&A=\left[\begin{array}{rr}0 & -1 \\\1 & 0 \\\\-1 & 2 \end{array}\right], B=\left[\begin{array}{rr}0.25 & -1 \\\0 & 0.5 \\\\-1 & 3\end{array}\right], \text { and } \\\&C=\left[\begin{array}{rr}1 & -1 \\\1 & 1 \\\\-1 & -1\end{array}\right].\end{aligned}$$ $$ A-C $$

Why is matrix addition associative?

T o u r i s m ~ i n ~ t h e ~ ' 9 0 s ~ T h e ~ f o l l o w i n g ~ t a b l e ~ g i v e s ~ t h e ~ n u m b e r ~ }\\\ &\text { of people (in thousands) who visited Australia and South }\\\ &\text { Africa in } 1998 .^{13} \end{aligned} You estimate that \(5 \%\) of all visitors to Australia and \(4 \%\) of all visitors to South Africa decide to settle there permanently. Take \(A\) to be the \(3 \times 2\) matrix whose entries are the 1998 tourism figures in the above table and take $$ B=\left[\begin{array}{l} 0.05 \\ 0.04 \end{array}\right] \text { and } C=\left[\begin{array}{ll} 0.05 & 0 \\ 0 & 0.04 \end{array}\right] $$ Compute the products \(A B\) and \(A C\). What do the entries in these matrices represent?
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