(a) Show that the elementary row operation matrix corresponding to the interchange of two rows i

(a) Show that the elementary row operation matrix...

Key Concepts

Orthogonal Matrix

An orthogonal matrix is a square matrix Q for which the transpose is equal to the inverse (Q? = Q?¹), meaning Q?Q = I, where I is the identity matrix. This property ensures that orthogonal matrices preserve the Euclidean norm and the inner product of vectors. In practical terms, they represent rotations and reflections in space, and hence are fundamental in various areas of linear algebra and geometry.

Proper versus Improper Orthogonal Matrices

Orthogonal matrices can be classified as proper or improper based on their determinant. A proper orthogonal matrix has a determinant of +1 and typically represents a pure rotation, whereas an improper orthogonal matrix has a determinant of -1 and generally corresponds to a reflection or a combination of a rotation and a reflection. In the case of the row interchange elementary matrix discussed, it is an example of an improper orthogonal matrix because its determinant is -1, thereby reversing the orientation of the vector space.

Elementary Row Operation Matrix

An elementary row operation matrix is a matrix that represents a single elementary row operation when multiplied to another matrix. These operations include row switching, row scaling, and adding a multiple of one row to another row. Such matrices are central in linear algebra as they help in processes like Gaussian elimination and matrix factorization. Their structure is simple and closely related to the identity matrix, modified only by the specific row operation performed.

Permutation Matrix

A permutation matrix is a square matrix obtained by permuting the rows or columns of an identity matrix. In the context of elementary row operations, the row interchange operation produces a permutation matrix. Permutation matrices are special because they have exactly one entry of 1 in each row and column (with all other entries 0), and they preserve the length of vectors under multiplication. They serve as the algebraic representation of rearranging the order of elements in vectors or matrices.

Orthogonality in Elementary Matrices

Not every elementary matrix is orthogonal. Orthogonality in elementary matrices occurs only when the elementary row operation preserves lengths and angles without scaling. Row permutation matrices (resulting from row swaps) are orthogonal because they only rearrange the rows of the identity matrix. Additionally, elementary matrices corresponding to multiplying a row by -1 (a special case of scaling by a unit modulus factor) are also orthogonal. However, the operation of adding a multiple of one row to another generally does not yield an orthogonal matrix since it can alter lengths and angles in a way that violates the orthogonal condition.

Transcript

00:01 So the question asks, if you're given a matrix a, and then you're given matrix b, which is a single elementary row operation away from a, is there a way to get back to matrix a? so to do that, let's look at a simple case.

00:26 We'll look at a single matrix, and then we'll gather a generalization from there.

00:33 So let's say we had the matrix.

00:37 Let's do 2, 3, 1, 4.

00:41 There are three elementary row operations, if you can recall them.

00:44 The first one is simply swapping two rows.

00:53 The second, the second is scaling any row by a constant number.

01:11 And the third is adding a scaled multiple of another row.

01:24 So that'll give you, for taking this matrix, that will give you three different matrix matrices so if we look at the first one if we just swap the rows you get the matrix one four two three if we scale by a multiple i will use the number two just for ease i'll do and i'll write it right above it so this one i just did our one swap r2 that one on the wrong side this one i'll do r1 multiplied by two we get four six one four and the last one if we do that one i'll do again two is for ease so r we do r2 goes to r1 plus r2 two two three two three and it's gonna be five ten and then the choice of two was just for ease since i already have them here so now let's look it going from here backwards.

03:00 So this one is actually, i think you could probably see it yourself.

03:04 If you just swap them back, you'll get the matrix you started with.

03:12 For this one, for this one, well, what if we just, so if we have r1 is now equal to two times the original r1, so we'll call it r1 minus, well if i just do, if i just divide it by two, i'm back to where i started.

03:40 So i will do r1 is one half r1...

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