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I understand and capable of doing basic matrix operations and related things.

But the thing I can't understand is,

How to construct a matrix from given data? How to know what to assign to columns, and what to rows?

Take this part from a textbook for example - enter image description here

How to know when to take row matrix, and when to take column matrix? I understand, by doing this the multiplication is made possible, but that's not a logical thinking.

About this particular example, my confusion is why the 'two parts' are assigned to columns in matrix $A$, then to rows in matrix $R$. To me it is inconsistency. Also later when $AR$ is done, it solves the problem, but it is not clear why $RA$ was not done. (Here $AR$=$RA$, but that's not the case every time.)

Also, apart from 'composition of two linear transformations', is there a 'day-to-day' meaning of matrix-multiplication? To clarify that, we consider $2\times3$ as $2$ added to zero $3$ times. Apart from this mathematical interpretation, we can also say that the product is the total number of something required if each of $2$ people are given $3$ of that thing. I was looking for similar 'easy' meaning for matrix multiplication.

Kawrno
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1 Answers1

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"apart from 'composition of two linear transformations', is there a 'day-to-day' meaning of matrix-multiplication?"

Not really. This form of matrix multiplication came from work on linear transformations; for example, if you were to transform $(x,y)$ through $(x'=1x-3y, y'=7x+5y)$ and then through $(x''=-1x'+2y',y''=-2x'+1y')$, and do all the necessary substitutions and simplifications to obtain equations for $(x,y)$ in terms of $(x'',y'')$, the rule for matrix multiplication comes out naturally, and I suppose because the rule was developed from this application, it is usefully applied to other areas involving linear equations.

"why the 'two parts' are assigned to columns in matrix A, then to rows in matrix R."

The rule for matrix multiplication says that the number of columns in the left matrix of the multiplication must be the same as the number of rows in the right matrix, and presumably the solution goes on to carry out the operation $AB$. Hence the assignment given in the solution. In fact, for this problem, $A$ could be assigned the values $0.09$ and $0.11$ with $B$ being given the other values, but, if we are carrying out the operation $AB$, then $A$ must still be a $1\times2$ matrix and $B$ a $2\times1$matrix.

So, to answer your original question, for problems like this, where matrix multiplication will be involved, the left matrix (of the multiplication) should have values assigned in columns, while the right matrix should have the values assigned to rows. And the reason is because of the way matrix multiplication is defined.

robert timmer-arends
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