Matrix product examples | Matrix transformations | Linear Algebra | Khan Academy

TL;DR

Matrix-matrix products are used to represent the composition of linear transformations and can only be defined when the number of columns in the first matrix is equal to the number of rows in the second matrix.

Transcript

In the last video we learned what it meant to take the product of two matrices. So if we have one matrix A, and it's an m by n matrix, and then we have some other matrix B, let's say that's an n by k matrix. And we've defined the product of A and B to be equal to-- And actually before I define the product, let me just write B out as just a collecti... Read More

Key Insights

• ✖️ Matrix-matrix products are defined by multiplying a matrix by each column vector of another matrix.
• #️⃣ The number of columns in the first matrix must equal the number of rows in the second matrix for the product to be well-defined.
• ❓ Matrix-matrix products can be used to represent the composition of linear transformations.
• #️⃣ The dimensions of the resulting product matrix are determined by the number of rows in the first matrix and the number of columns in the second matrix.
• 🔙 The product of matrix A and B is not always equal to the product of matrix B and A, and sometimes the latter is not even defined.

Questions & Answers

Q: What is the definition of a matrix-matrix product?

A matrix-matrix product is obtained by multiplying a matrix A by each column vector of another matrix B.

Q: When can a matrix-matrix product be defined?

A matrix-matrix product is only well-defined when the number of columns in the first matrix is equal to the number of rows in the second matrix.

Q: What is the significance of the number of columns and rows in matrix-matrix products?

The number of columns in the first matrix and the number of rows in the second matrix determine whether the product is well-defined and the dimensions of the resulting product matrix.

Q: How are matrix-matrix products related to linear transformations?

Matrix-matrix products can represent the composition of linear transformations, with each matrix being the transformation matrix of an individual transformation.

Summary & Key Takeaways

• The product of two matrices A and B is defined as the matrix A multiplied by each column vector of B.

• Matrix-matrix products are only well-defined when the number of columns in A is equal to the number of rows in B.

• Matrix-matrix products can be used to represent the composition of linear transformations and are equivalent to the product of the transformation matrices of the individual transformations.