Step by step
The question
Question: multiply these two matrices.
[2, 0, −1; 1, 3, 2] × [1, 2; 0, −1; 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.
= [2 × 1 + 0 × 0 + −1 × 4, 2 × 2 + 0 × (−1) + −1 × 1; 1 × 1 + 3 × 0 + 2 × 4, 1 × 2 + 3 × (−1) + 2 × 1]
Careful!
Always a row of the first matrix with a column of the second.
Never the other way.
Careful: row of the first × column of the second
3 Work out each entry
First we work out every times.
= [2 + 0 + (−4), 4 + 0 + (−1); 1 + 0 + 8, 2 + (−3) + 2]
Then we add.
The answer: rows −2, 3 and 9, 1.
= [−2, 3; 9, 1]
4 Check
Let's check each entry, adding its products the other way round.
−4 + 0 + 2 = −2, −1 + 0 + 4 = 3
8 + 0 + 1 = 9, 2 + (−3) + 2 = 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
= [−2, 3; 9, 1]