Step by step
The question
Question: multiply these two matrices.
[−1, 2; 0, 3] × [2, −3; 4, 1]
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 × 4, −1 × (−3) + 2 × 1; 0 × 2 + 3 × 4, 0 × (−3) + 3 × 1]
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 + 8, 3 + 2; 0 + 12, 0 + 3]
Then we add.
The answer: rows 6, 5 and 12, 3.
= [6, 5; 12, 3]
4 Check
Let's check each entry, adding its products the other way round.
8 + (−2) = 6, 2 + 3 = 5
12 + 0 = 12, 3 + 0 = 3
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
= [6, 5; 12, 3]