Step by step
The question
Question: multiply these two matrices.
[4, 0; −2, 1; 3, 5] × [1, −1; 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.
= [4 × 1 + 0 × 2, 4 × (−1) + 0 × 3; −2 × 1 + 1 × 2, −2 × (−1) + 1 × 3; 3 × 1 + 5 × 2, 3 × (−1) + 5 × 3]
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.
= [4 + 0, −4 + 0; −2 + 2, 2 + 3; 3 + 10, −3 + 15]
Then we add.
The answer: rows 4, −4, 0, 5 and 13, 12.
= [4, −4; 0, 5; 13, 12]
4 Check
Let's check each entry, adding its products the other way round.
0 + 4 = 4, 0 + (−4) = −4
2 + (−2) = 0, 3 + 2 = 5
10 + 3 = 13, 15 + (−3) = 12
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, −4; 0, 5; 13, 12]