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