I'm studying linear algebra using the online MIT course, and in the third lecture, the professor showed us 5 ways to multiply matrices, they can be found here:MIT Linear Algebra, Lecture 3: Matrix Multiplication and Inverse Matrices
And he said that all the 5 ways are important, which means I have to master them all. Can you explain to me what's so important in learning matrix multiplication in five different ways?
Best Answer
It's not really five different ways, but five ways to use the same definition. Sometimes viewing something from a different persepective makes things easier.
Let's give an example. Let $A$ and $X$ be compatible matrices and write $X_i$ for the $i$-th column of $X$. Then one the five ways says that the $i$-th column of $AX$ is $AX_i$. We will use this property to prove the following:
Let $A$ be a square matrix and suppose that $AX=B$ has a solution for each column $B$. Then $A$ is invertible.
Indeed, define $B_i$ as the column with zeroes everywhere and a one on the $i$-th spot. Then there exists an $X_i$ such that $AX_i=B_i$. Define $X$ as the matrix with the $X_i$'s in the columns, then $AX=Id$ by the property above. Thus $A$ has a right inverse which is sufficient since $A$ is square.