Step by step
The question
Question: multiply these two matrices.
[3, 1; 1, 2] × [1, 1; 2, 0]
1 Check the sizes
2 Row times column
Each entry of the answer is one row of the first matrix, times one column of the second.
= [3 × 1 + 1 × 2, 3 × 1 + 1 × 0; 1 × 1 + 2 × 2, 1 × 1 + 2 × 0]
Careful!
We do not just multiply the entries in the same place.
Each entry is a whole row times a whole column.
Careful: not entry × entry ✗
3 Work out each entry
First we work out every times.
= [3 + 2, 3 + 0; 1 + 4, 1 + 0]
Then we add.
The answer: rows 5, 3 and 5, 1.
= [5, 3; 5, 1]
4 Check
Let's check each entry, adding its products the other way round.
2 + 3 = 5, 0 + 3 = 3
4 + 1 = 5, 0 + 1 = 1
Every entry matches.
So the answer is right!
Let's look back.
First: check the sizes.
Then: row times column.
Then: work out each entry.
Last: check.
Answer
= [5, 3; 5, 1]