Science:Math Exam Resources/Courses/MATH152/April 2017/Question A 23

MATH152 April 2017

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[hide] Question A 23
Let $A$ and $B$ be matrices and let ${\mathbf {x}}$ be a vector. If $(A{\mathbf {x}})^{T}B$ is an $4\times 7$ matrix, is ${\mathbf {x}}$ a row vector or a column vector, and how many entries does it have? Justify your answer.

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[show] Hint
For matrix multiplication the number of columns of the first matrix must be the same as the number of rows of the second one.

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[show] Solution

We can only multiply 2 matrices $X\times Y$ if the number of columns of $X$ is the same as the number of rows of $Y$. Thus if the product $(A{\mathbf {x}})^{T}B$ is a $4\times 7$ matrix, then we know $(A{\mathbf {x}})^{T}$ has size $4\times m$ and $B$ has size $m\times 7$, for some positive integer $m$. If the transposed matrix $(A{\mathbf {x}})^{T}$ is size $4\times m$, then the original matrix $A{\mathbf {x}}$ must be size $m\times 4$. This implies that $A$ is of size $m\times n$ and ${\mathbf {x}}$ is of size $n\times 4$ for some positive integers $m,n.$ Now, if ${\mathbf {x}}$ were a column vector, then $A{\mathbf {x}}$ would also be a column vector, of size $l\times 1$ for some $l$. However, since $4\neq 1$, we know that ${\mathbf {x}}$ must be a row vector. Finally, a row vector ${\mathbf {x}}$ is of size $1\times k$ for some $k$, and we know from our earlier reasoning ${\mathbf {x}}$ is also of size $n\times 4$. Thus, for our matrix operations to agree, we must have that $\color {blue}{\mathbf {x}}{\text{ is a }}1\times 4{\text{ row vector, with }}4{\text{ entries}}.$