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