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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?

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    Well IMO, the $2,3,4$ methods are really redundant. After you multiply matrix with column or row, you essentially do what you did in the first method. – 2017-02-20

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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.