# Composing 3x3 matrices | Matrices | Precalculus | Khan Academy | Summary and Q&A

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March 21, 2021
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Composing 3x3 matrices | Matrices | Precalculus | Khan Academy

## TL;DR

This video explains how to compose two three by three matrices as transformations in three-dimensional space.

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### Q: How can we represent the composition of two matrices as transformations?

The composition of two matrices can be represented by applying the second matrix first and then applying the first matrix. This can be thought of as transforming each column of the second matrix with the first matrix.

### Q: How is the middle column of the resulting matrix determined in the composition of A of B?

The middle column of the resulting matrix is determined by applying matrix A to the middle column of matrix B. Each entry in the middle column is transformed using the corresponding basis vectors under matrix A's transformation.

### Q: What are the basis vectors used in the transformation?

Instead of using the standard unit vectors, the basis vectors used in the transformation are the images of the standard unit vectors under the transformation of matrix A.

### Q: What is the process for finding the middle column of the resulting matrix?

To find the middle column, each entry is calculated by multiplying the corresponding entry from the middle column of matrix B with the transformed basis vectors under matrix A. The resulting entries are then added up to obtain the final column.

## Summary & Key Takeaways

• The video demonstrates how to compose two matrices A and B as transformations.

• To compose A of B, each column of matrix B is transformed using matrix A.

• The middle column of the resulting matrix is determined by applying matrix A to the middle column of matrix B.