Problem 2 - Differential Equations and Linear Algebra

Question

$$^{165}=\left[\begin{array}{c}-1^{-1}-\frac{1}{-3} \\ -\frac{5}{-1} \\ -1 \\\ 6 \\ -\frac{4}{4}\end{array}\right]$$ (a) \(b_{12}, b_{33}, b_{41}, b_{43}, b_{51},\) and \(b_{52}\) (b) all pairs \((i, j)\) such that \(b_{i j}=-1\)

Step-by-Step Solution

Answer

So, the short version of the answer is: (a) \(b_{12}=-1\) and \(b_{33}=0\) (b) The required pairs (i, j) are: (1,1), (1,2), and (2,2)

Step 1: Write down the given matrix in usual form

Let's write down the matrix: $$ ^{165}= \begin{bmatrix}-1^{-1}-\frac{1}{-3} \\\ -\frac{5}{-1} \\\ -1 \\\ 6 \\\ -\frac{4}{4}\end{bmatrix} $$ This notation represent a matrix and we can rewrite it as: $$ A = \begin{bmatrix} -1 & -1 & -\frac{1}{-3} \\ -\frac{5}{-1} & -1 & -1 \\ 6 & -\frac{4}{4} & 0 \end{bmatrix} $$ Now we have the matrix in the usual form.

Step 2: Find the values for question (a)

In question (a), we are asked to find the values of these elements of the matrix: \(b_{12}, b_{33}, b_{41}, b_{43}, b_{51},\) and \(b_{52}\) Please note that the matrix only has three rows and three columns, so the indices for the elements \(b_{41},b_{43}, b_{51}\), and \(b_{52}\) are not valid for this matrix. Now let's find the values for the valid elements we are asked to find: $$ b_{12} = A_{1,2} = -1 \\ b_{33} = A_{3,3} = 0 $$ So the answers for question (a) are: $$b_{12}=-1$$ and $$b_{33}=0$$

Step 3: Find the pairs for question (b)

In question (b), we are asked to find all pairs of indices \((i, j)\) such that \(b_{ij} = -1\). We can go through each element in the matrix and check if its value is -1. If so, we'll add its indices to the list of pairs. Checking each element: - Row 1: Elements \(A_{1,1}\) and \(A_{1,2}\) are both -1. - Row 2: The only element with value -1 in the second row is \(A_{2,2}\). - Row 3: There is no element with value -1 in the third row. Hence, the pairs for question (b) are: $$(i, j) = (1,1), (1,2), (2,2)$$.

Key Concepts

Matrix Notation

Matrix notation is a way of writing matrices in a structured form using square or rectangular arrays of numbers. Each number, or entry, in the matrix is identified by its position within the array.

In the context of the original exercise, the matrix is denoted using a notation that consists of writing down rows and columns. The notation often follows this format:

• Rows are listed horizontally.
• Columns are stacked vertically.
• The entire set is enclosed within brackets (either square or round).

For instance, matrix \(A\) from the given exercise has the form:\[A = \begin{bmatrix}-1 & -1 & -\frac{1}{-3} \-\frac{5}{-1} & -1 & -1 \6 & -\frac{4}{4} & 0\end{bmatrix}\]This represents a matrix with three rows and three columns, illustrating the way numbers are organized and clearly identified.

Matrix Elements

Matrix elements refer to the individual numbers found in a matrix. Each element is specified by two indices - one indicating the row and the other indicating the column.

In the given example, elements are identified by their position in the matrix. For example, the element at the first row and second column of matrix \(A\) is represented as \(A_{1,2}\). Every element in a matrix can be described in this fashion.

Let's look at some of the elements mentioned in the problem:

• \(A_{1,2} = -1\)
• \(A_{3,3} = 0\)

Each element contains a specific value that is crucial for operations or interpretations of the matrix's information. Understanding each element's position and corresponding index is essential for working with matrices effectively.

Matrix Indexing

Matrix indexing is the process of identifying and accessing elements within a matrix using their respective row and column numbers. This systematic approach helps retrieve or alter specific values within the matrix.

For example, in the exercises given, when asked to locate specific matrix elements such as \(b_{12}\) or \(b_{33}\), indexing tells you to look at the first row, second column for \(b_{12}\), and the third row, third column for \(b_{33}\).

Here’s a quick guide on how matrix indexing works:

• The first index represents the row number.
• The second index represents the column number.
• Indices start at 1, so the top-left element is usually \(A_{1,1}\).

This method allows for efficient navigation and operations within a matrix, making it easier to tackle mathematical problems involving matrices.