Tell whether the product QP is defined for P3x7 and Q5x3. If yes, give the product's dimensions as the following example: 4x3. If no, write no in the box.

Question

agent0smith · Answer

How do you tell if the product of two matrices is defined?

campayne · Answer

ive not in the slightest clue

holsteremission · Answer

Think about how the dot product is defined. Given two vectors $\mathbf x,\mathbf y\in\mathbb R^n$ with components $x_1,\ldots,x_n$ and $y_1,\ldots,y_n$, the dot product is given as the sum of the products of corresponding components, i.e. $\mathbf x^\intercal\mathbf y=\mathbf x\cdot\mathbf y=x_1y_1+\cdots+x_ny_n$.

So if you write this product in matrix form (so $\mathbf x\in\mathbb R^{1\times n}$ and $\mathbf y\in\mathbb R^{n\times1}$), you have
$$\mathbf x^\intercal\mathbf y=\begin{bmatrix}x_1&\cdots&x_n\end{bmatrix}\cdot\begin{bmatrix}y_1\\vdots\y_n\end{bmatrix}$$then it should be clear that the number of columns of the first matrix must match the number of rows in the second matrix.

What this means is that given two matrices $\mathbf X\in\mathbb R^{a\times b}$ and $\mathbf Y\in\mathbb R^{c\times d}$, the product $\mathbf{XY}$ can only exist if $b=c$.

Now suppose $\mathbf X$ consists of two rows, but the same number of columns as $\mathbf x^\intercal$, so that $\mathbf X\in\mathbb R^{2\times n}$. The product $\mathbf {Xy}$ is then, according to the rules of matrix multiplication,
$$\mathbf {Xy}=\begin{bmatrix}x_{1,1}&\cdots&x_{1,n}\x_{2,1}&\cdots&x_{2,n}\end{bmatrix}\begin{bmatrix}y_1\\vdots\y_n\end{bmatrix}=\begin{bmatrix}x_{1,1}y_1+\cdots+x_{1,n}y_n\x_{2,1}y_1+\cdots+x_{2,n}y_n\end{bmatrix}$$so $\mathbf{Xy}\in\mathbb R^{2\times n}$.

Extrapolating, this means that given two appropriately sized matrices $\mathbf X\in\mathbb R^{a\times b}$ and $\mathbf Y\in\mathbb R^{b\times c}$, the product $\mathbf{XY}\in\mathbb R^{a\times c}$.

holsteremission · Answer

You know that $\mathbf Q\in\mathbb R^{5\times3}$ and that $\mathbf P\in\mathbb R^{3\times7}$, so the number of columns of $\mathbf Q$ matches the number of rows of $\mathbf P$. This means the product $\mathbf{QP}$ does exist.

What are the dimensions of this product?

holsteremission · Answer

Correction to my first post: $\mathbf{Xy}\in\mathbb R^{2\times1}$, not $\mathbb R^{2\times n}$

agent0smith · Answer

If you google what i asked "How do you tell if the product of two matrices is defined?" it tells you how to tell.

campayne · Answer

i got 5 x 7

holsteremission · Answer

Correct

campayne · Answer

but i got it wrong on my thing