Step by step
The question
Question: multiply these two matrices.
[1, 0; 0, 1] × [7, 2; −3, 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.
= [1 × 7 + 0 × (−3), 1 × 2 + 0 × 5; 0 × 7 + 1 × (−3), 0 × 2 + 1 × 5]
Careful!
We do not just multiply the entries in the same place.
Each entry is a whole row times a whole column.
Careful: not entry × entry ✗
3 Work out each entry
First we work out every times.
= [7 + 0, 2 + 0; 0 + (−3), 0 + 5]
Then we add.
The answer: rows 7, 2 and −3, 5.
= [7, 2; −3, 5]
4 Check
Let's check each entry, adding its products the other way round.
0 + 7 = 7, 0 + 2 = 2
−3 + 0 = −3, 5 + 0 = 5
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
= [7, 2; −3, 5]