Question

Rules of matrices

I don't understand this multiply matrices thing, what is required for me to be able to do this?

Answer 1

for the multiplication to be "legal"?

Answer 2

Yeah I guess? Particularly for uneven matrices like say A = 5x4 and B = 4x3

That sort of thing.

Answer 3

stolen from wikipedia but the product AB is defined only if the number of columns in A is equal to the number of rows in B

Answer 4

for example if A was 3x2 and B was 2x3 it'd be defined
if A was 3x3 and B was 2x2 it wouldn't be defined

Answer 5

So 5x4 and 4x3 wouldn't work?

Answer 6

look at the inner most numbers
the way matrices are expressed it's rowx col

Answer 7

so
\(axb * cxd\) is defined when b=c

Answer 8

this is part of why matrix multiplication also isn't commutative
for example,
2x2 * 1x2 isn't defined
but
2x1 * 2x2 is defined

Answer 9

and the outer numbers tells you the size of the product matrix. So a 2x1 * 2x2 multiplication will result in a 2x2 matrix and 5x4 * 4x3 gives a 5x3 matrix

Answer 10

The number of columns of the left matrix must equal the rows of the right matrix.

Answer 11

Matrices come from systems of equations. For example: \[
ax+by = e\\
cx+dy=f
\]We can write this a vector equation: \[
\begin{bmatrix}a\\c\end{bmatrix}x+\begin{bmatrix}b\\d\end{bmatrix}y=\begin{bmatrix}e\\f\end{bmatrix}
\]However, if we want to treat the variables \(x\) and \(y\) as components of a vector, then we need matrix multiplication to be defined to work this way:\[
\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}ax+by\\cx+dy\end{bmatrix}
\]This way we can say: \[
\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}e\\f \end{bmatrix}
\]

Answer 12

Okay... but 5x4 and 4x3 will work.

Answer 13

Matrices are denoted with rows by columns.

If the left matrix has 4 columns, and the right matrix has 4 rows, then it should work.

Answer 14

And we have 4 columns and 4 rows.

Answer 15

Which means this is possible, right?