# Proving two simple matrix product properties

• TheSodesa
In summary, the conversation discusses proving the matrix product of two matrices, A and B, where B is represented as a column vector and A is represented with the help of its row vectors. The homework equations for calculating the matrix product are provided. The attempt at a solution involves calculating C = AB and D = [A b1 ... A bm], and showing that Dij = Cij for any i,j.
TheSodesa

## Homework Statement

Let ##A## be an n × p matrix and ##B## be an p × m matrix with the following column vector representation,

$$B = \begin{bmatrix} b_1 , & b_2, & ... & ,b_m \end{bmatrix}$$

Prove that
$$AB = \begin{bmatrix} Ab_1 , & Ab_2, & ... & , Ab_m \end{bmatrix}$$

If ##A## is represented with help of its row vectors, prove that

$$AB = \begin{bmatrix} a^{T}_{1}\\ \vdots\\ a_{n}^{T} \end{bmatrix} B = \begin{bmatrix} a^{T}_{1} B\\ \vdots\\ a^{T}_{n} B \end{bmatrix}$$

## Homework Equations

The matrix product:
If ##A## is an ##m\times p## matrix and ##B## is a ##p\times n## matrix, then

AB = C = (c_{ij})_{m \times n} = (\sum_{k=1}^{p} a_{ik}b_{kj})_{m \times n}

## The Attempt at a Solution

For starters what does proving in this context mean? Should I simply write out the matrix

$$C = \begin{bmatrix} \sum_{k=1}^{p} a_{1k} b_{k1} & \cdots & \sum_{k=1}^{p} a_{1k} b_{km}\\ \vdots & \ddots & \vdots\\ \sum_{k=1}^{p} a_{mk} b_{k1} & \cdots &\sum_{k=1}^{p} a_{mk} b_{km} \end{bmatrix}$$
and conclude that each column is essentially ##Ab_{l}## where ##1 < l < m##, since each element of the matrix is the dot (inner) product of a row in ##A## and a column in ##B##?

Last edited:
Oops. This

$$C = \begin{bmatrix} \sum_{k=1}^{p} a_{1k} b_{k1} & \cdots & \sum_{k=1}^{p} a_{1k} b_{km}\\ \vdots & \ddots & \vdots\\ \sum_{k=1}^{p} a_{mk} b_{k1} & \cdots &\sum_{k=1}^{p} a_{mk} b_{km} \end{bmatrix}$$

should be this
$$C = \begin{bmatrix} \sum_{k=1}^{p} a_{1k} b_{k1} & \cdots & \sum_{k=1}^{p} a_{1k} b_{km}\\ \vdots & \ddots & \vdots\\ \sum_{k=1}^{p} a_{nk} b_{k1} & \cdots &\sum_{k=1}^{p} a_{mk} b_{km} \end{bmatrix}$$
since it's an ##n \times m## matrix, not an ##m \times m## matrix.

1. Calculate C = AB.
2. Calculate D = [A b1 ... A bm]
3. Show Dij = Cij for any i,j

## 1. How do I prove that matrix multiplication is associative?

To prove that matrix multiplication is associative, you must show that (AB)C = A(BC) for any three matrices A, B, and C. This can be done by expanding the left and right sides of the equation and showing that they are equal.

## 2. What is the identity property of matrix multiplication?

The identity property of matrix multiplication states that the product of any matrix and the identity matrix is equal to the original matrix. This can be shown by multiplying any matrix A by the identity matrix I and proving that the result is equal to A.

## 3. How can I prove the distributive property of matrix multiplication?

The distributive property of matrix multiplication states that A(B+C) = AB + AC for any matrices A, B, and C. This can be proven by expanding the left and right sides of the equation and showing that they are equal.

## 4. Is matrix multiplication commutative?

No, matrix multiplication is not commutative. This means that for two matrices A and B, AB is not necessarily equal to BA. A simple counterexample would be to consider two different matrices A and B with different dimensions, which would result in different products AB and BA.

## 5. Can I use the properties of matrix multiplication to simplify expressions?

Yes, the properties of matrix multiplication can be used to simplify expressions by rearranging the order of operations. This can be particularly helpful when dealing with large matrices or complicated expressions.

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