Step by step
The question
Question: multiply these two matrices.
[2, −1; 3, 4] × [1, 5; −2, 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.
= [2 × 1 + −1 × (−2), 2 × 5 + −1 × 3; 3 × 1 + 4 × (−2), 3 × 5 + 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, 10 + (−3); 3 + (−8), 15 + 12]
Then we add.
The answer: rows 4, 7 and −5, 27.
= [4, 7; −5, 27]
4 Check
Let's check each entry, adding its products the other way round.
2 + 2 = 4, −3 + 10 = 7
−8 + 3 = −5, 12 + 15 = 27
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, 7; −5, 27]