Find a 4 ×4 matrix A=[ai j] whose entries satisfy the stated condition. (a) ai j=i+j (b) ai j=i^

Find a 4 ×4 matrix A=[ai j] whose entries satisfy the stated...

Key Concepts

Vandermonde-Type Structure

A Vandermonde matrix is characterized by having each row (or column) constructed from successive powers of a variable, typically following the form a_ij = x_i^(j-1). This type of structure, or its variants, appears in situations that involve polynomial interpolation or determinant evaluations, and it is particularly notable for its distinctive algebraic properties.

Matrix Notation and Indexing

Matrices are arranged in rows and columns, and each entry is typically denoted by a subscript that indicates its position within the array (i.e., a_ij represents the entry in the i-th row and j-th column). This notation allows one to define matrix entries using functions of the indices, making it possible to systematically specify all entries of a matrix.

Entry-Wise or Index-Based Definitions

Defining a matrix via an explicit formula for its entries, such as a_ij = f(i, j), is a common technique in linear algebra. This approach uses the row and column indices to generate the value of each entry according to a specified rule, ensuring that the structure of the matrix is driven directly by its indexing.

Piecewise-Defined Matrices and Band Structure

In some matrices, entries are defined using different formulas based on the relationship between the indices, such as conditions involving the difference |i-j|. This results in a matrix with different regions or bands that follow different patterns, which is typical in the study of band matrices, sparse matrices, or matrices with specific combinatorial properties.

Transcript

00:01 In this problem, we are constructing four by four matrices to satisfy some sort of condition that we're given.

00:09 And we have to do this three different times.

00:12 So what we need to pay attention in this problem is the indices that we're given.

00:17 So for example, in this first part a, we define a matrix sub ij, where i and j are the number of our rows and columns.

00:26 So that's just something that we need to keep track of because that's going to tell us.

00:31 Essentially how to construct these matrices.

00:35 So for part a, we're told that a sub ij is equal to i plus j.

00:41 So what does that look like in our matrices? and i will read them going down a column.

00:47 So this first column will have 1 plus 1.

00:50 So this is essentially our first column plus our first row or vice versa.

00:57 So down this column we'd have 1 plus 1, 2 plus 1, 3 plus 1, 3 plus 3.

01:01 1, 4 plus 1, where this first number is the number of our row and the second number is the number of our column.

01:09 And then we have this pattern continue.

01:11 So our second column would be 1 plus 2, 2 plus 2, 3 plus 2, 4 plus 2, where when we add 2 each time, that's the number of our column.

01:21 So this repeats until we get up to 4 columns.

01:25 And we'll find that our final 4x, when i'm going to read this again by column, would be 2.

01:31 Two, three, four, five, five, six, four, five, six, seven, five, six, seven, eight.

01:39 And that is our four by four matrix.

01:46 Now, for part b, i think i wrote this a little bit incorrectly.

01:50 I think i rewrote a, but part b, i believe, was a sub ij raised to ij...

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