Find $(AB)^T$: Calculate Matrix Product & Transpose

In summary, we are given matrices A and B and are asked to find the transpose of their product AB. We first find the product AB by multiplying the corresponding elements and adding them together. Then, we transpose the resulting matrix by reflecting its rows across its main diagonal. This results in a new matrix where the rows of the original matrix become the columns of the transposed matrix.
  • #1
karush
Gold Member
MHB
3,269
5
Let
$A=\left[\begin{array}{c}1 & 2 & -3 \\ 2 & 0 & -1 \end{array}\right] \textit { and }
B=\left[\begin{array}{c}3&2 \\ 1 & -1 \\ 0 & 2 \end{array}\right]$
Find $(AB)^T$$AB=\left[ \begin{array}{cc}(1\cdot 3)+(2\cdot1)+(-3\cdot0) & (1\cdot2)+(2\cdot-1)+(-3\cdot2) \\
(2\cdot3)+(0\cdot1)+ (-1\cdot0) & (2\cdot2)+(0\cdot-1)+(-1\cdot2) \end{array} \right]=\left[\begin{array}{c}5 & -6 \\ 6 & 2 \end{array}\right]$
then transpose I think this is just a diagonal reflection?
$\left[\begin{array}{c}5 & -6 \\ 6 & 2 \end{array}\right]^T=\left[\begin{array}{c}5 & 6 \\ -6 & 2 \end{array}\right]$
ok think this is correct but would like comments .. if any
 
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  • #2
Looks good to me!
However, I wouldn't refer to it as the "diagonal reflection", but that is only my opinion. I only say this because a matrix need not be a square matrix to take the transpose.

I think of it as, the rows of the original matrix become the columns of the transposed matrix.

Just as an example...

$M =\left[\begin{array}{c}2 & 3 & -1 \\ 3 & 1 & 0 \end{array}\right] $
$M^T =\left[\begin{array}{c}2 & 3 \\ 3 & 1 \\ -1 & 0 \end{array}\right] $
 

Related to Find $(AB)^T$: Calculate Matrix Product & Transpose

1. What is the purpose of finding the transpose of a matrix product?

The transpose of a matrix product is useful in various mathematical applications, such as solving systems of equations and finding eigenvalues and eigenvectors. It also helps in simplifying calculations and making them more efficient.

2. How do you calculate the transpose of a matrix product?

To calculate the transpose of a matrix product, first find the product of the two matrices. Then, swap the rows and columns of the resulting matrix to get the transpose. This can be done by reflecting the elements along the main diagonal.

3. Can the transpose of a matrix product be calculated in any order?

No, the transpose of a matrix product is not commutative, meaning that the order in which the matrices are multiplied matters. In general, (AB)^T is not equal to (A^T)(B^T). However, there are special cases where this may hold true, such as when one of the matrices is an identity matrix.

4. What is the difference between the transpose of a matrix product and the product of transposes?

The transpose of a matrix product is calculated by finding the product of two matrices and then taking the transpose of the resulting matrix. On the other hand, the product of transposes is calculated by taking the transpose of each individual matrix and then finding their product. The resulting matrices may not be the same in these two cases.

5. Are there any specific properties of the transpose of a matrix product?

Yes, there are a few properties of the transpose of a matrix product that can be useful in calculations. For example, (AB)^T = B^T A^T and (kA)^T = kA^T, where k is a scalar. Additionally, if a matrix is symmetric, its transpose is equal to itself, i.e. A^T = A.

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