Adding Matrices
Matrices add the same friendly way vectors do — entry by entry. Line them up
and add the numbers that share a position:
\begin{bmatrix} a & b \\ c & d \end{bmatrix} + \begin{bmatrix} e & f \\ g & h \end{bmatrix} = \begin{bmatrix} a+e & b+f \\ c+g & d+h \end{bmatrix}.
Because the entries must line up, you can only add matrices of the same size —
a 2\times 2 to a 2\times 2, never a
2\times 2 to a 2\times 3. Subtraction and
scalar multiplication work entry-wise too: 3A just triples every
number in A.
One cell at a time
Step through the four positions below. Each move adds the two highlighted numbers to fill the
matching cell of the answer. That's the whole operation — no surprises.
The same old rules
Matrix addition is commutative and associative, and there's a
zero matrix (all entries 0) that changes nothing.
These are the exact rules numbers and vectors already follow — addition is the easy operation.
Multiplication, coming up, is where matrices get interesting and a little rebellious.