Step by step
The question
Question: multiply these two matrices.
[1, 2, 3; 0, 1, 4] × [1; −1; 2]
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 × 1 + 2 × (−1) + 3 × 2; 0 × 1 + 1 × (−1) + 4 × 2]
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.
= [1 + (−2) + 6; 0 + (−1) + 8]
Then we add.
The answer: the column 5, 7.
= [5; 7]
4 Check
Let's check each entry, adding its products the other way round.
6 + (−2) + 1 = 5
8 + (−1) + 0 = 7
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
= [5; 7]