Step by step
The question
Question: multiply these two matrices.
[5, −2; 1, 0] × [3, 1; 2, −4]
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.
= [5 × 3 + −2 × 2, 5 × 1 + −2 × (−4); 1 × 3 + 0 × 2, 1 × 1 + 0 × (−4)]
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.
= [15 + (−4), 5 + 8; 3 + 0, 1 + 0]
Then we add.
The answer: rows 11, 13 and 3, 1.
= [11, 13; 3, 1]
4 Check
Let's check each entry, adding its products the other way round.
−4 + 15 = 11, 8 + 5 = 13
0 + 3 = 3, 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
= [11, 13; 3, 1]