Transpose of a matrix product | Matrix transformations | Linear Algebra | Khan Academy

TL;DR

Matrix transposes and products have a relationship where transposing the product of two matrices is equal to the product of their transposes in reverse order.

Transcript

I've got a handful of matrices here. I have the matrix A that's an m by n matrix. You can see it has n columns and m rows. And actually let me throw in one entry there. It might be useful. This is the jth column. So the row is going to look like this-- amj. That's that entry right there. And then I have matrix B defined similarly, but instead of be... Read More

Key Insights

• 🎅 Matrix C, obtained by multiplying matrix A and B, has dimensions of m by m.
• 🤨 The entry cij in matrix C is the dot product of the ith row in A with the jth column in B.
• 🔙 Matrix D, obtained by multiplying the transposes of A and B, has the same dimensions as A and B.
• 🤨 The entry dji in matrix D is the dot product of the jth row in B transpose with the ith column in A transpose.
• ⌚ C transpose is equal to D, which can also be written as A times B transpose is equal to B transpose times A transpose.

Questions & Answers

Q: What are the dimensions of matrix C when A and B are multiplied?

Matrix C will have dimensions of m by m.

Q: How are the entries of matrix C calculated?

The entry cij in matrix C is obtained by taking the dot product of the ith row in A with the jth column in B.

Q: What are the dimensions of matrix D when B transpose and A transpose are multiplied?

Matrix D will have the same dimensions as A and B, which is m by n.

Q: How is the general entry dji in matrix D calculated?

The entry dji in matrix D is calculated by taking the dot product of the jth row in B transpose with the ith column in A transpose.

Q: What relationship is found between C and D?

It is discovered that C transpose is equal to D, meaning that the transpose of the product of A and B is equal to the product of the transpose of B and the transpose of A.

Summary & Key Takeaways

• The video discusses the concepts of matrix transposes and products, explaining their dimensions and how they are calculated.

• Two matrices, A and B, are defined along with their transposes. The dimensions of the resulting matrices after multiplication are also explained.

• The video demonstrates how to calculate specific entries in the matrices C and D by taking dot products of rows and columns. It is then shown that C transpose is equal to D.