Let A=[ai j] be an n ×n matrix. Determine whether A is symmetric. (a) ai j=i^2+j^2 (b) ai j=i^2-

step 1 Okay, so we have a matrix A, which is an n by n square matrix. And the elements within that matr

Let A=[ai j] be an n ×n matrix. Determine whether A is...

Key Concepts

Properties of Algebraic Expressions in Matrix Elements

Algebraic properties, such as commutativity of addition and differences in exponentiation or subtraction operations, are crucial when evaluating the symmetry of a matrix. For instance, symmetric expressions often involve sums that do not change when indices are swapped, whereas differences or non-commutative operations may not preserve equality under index interchange.

Matrix Entry Dependence on Indices

When matrix elements are defined by functions of the row and column indices (i and j), determining symmetry involves checking whether the function is invariant under swapping these indices. If the function yields the same result when the roles of i and j are interchanged, the resulting matrix will be symmetric.

Symmetric Matrix Definition

A symmetric matrix is one where the element in the i-th row and j-th column is equal to the element in the j-th row and i-th column for all indices, meaning the matrix is equal to its transpose. This property is fundamental in linear algebra and affects many theoretical and practical applications, such as eigenvalue computation and quadratic forms.

Transcript

00:01 Okay, so we have a matrix a, which is an n by n square matrix.

00:05 And the elements within that matrix, we can designate by a sub i, j, or i is the row, j is the column.

00:14 Now, the question is, if we have four different situations here, and we want to know whether a is symmetric.

00:21 Well, if a is symmetric, that means that i can interchange the rows and the columns.

00:28 So the element in the i row j column is going to be the same as the element in the j row i column for, you know, and just as a reminder, what that looks like.

00:40 That means if i have, let me just put, you know, three by three, four, six, seven along the top.

00:45 I have six in the first row second column, which means i would also have to have six in the second row first column.

00:53 If i have seven out here, i would have to have seven here.

00:56 That is what makes this symmetric.

00:59 So let's take a look.

01:00 I'm going to look at each of these cases in turn.

01:02 I'm going to look at the i -jeth element and compare it to the j -i -th element.

01:10 That sounds kind of funny, but i'm going to switch the i's and the js.

01:15 If it's equally valid, no matter which way i look at it, this is symmetric.

01:19 If it's different, it's not.

01:21 So let's take our first case.

01:23 The first one says a sub i -j.

01:27 Is going to be, that element is going to be i squared plus j squared.

01:33 Well, what if i transpose or transpose the i and the j? now i've got j squared plus i squared.

01:41 Well, yes, these are the same...

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