Step by step
The question
Question: multiply these two matrices.
[1, 2; 3, 4] × [2, 0; 1, 3]
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.
= [1 × 2 + 2 × 1, 1 × 0 + 2 × 3; 3 × 2 + 4 × 1, 3 × 0 + 4 × 3]
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.
= [2 + 2, 0 + 6; 6 + 4, 0 + 12]
Then we add.
The answer: rows 4, 6 and 10, 12.
= [4, 6; 10, 12]
4 Check
Let's check each entry, adding its products the other way round.
2 + 2 = 4, 6 + 0 = 6
4 + 6 = 10, 12 + 0 = 12
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
= [4, 6; 10, 12]