Step by step
The question
Question: find the determinant by row operations of this matrix.
D = |1, 2, 3; 0, 1, 4; 2, 5, 9|
We call the rows R 1, R 2 and R 3.
1 The rules
Row moves change D in a known way.
A swap changes its sign.
A number taken out of a row goes in front.
Adding a multiple of one row to another changes nothing.
And with 0s under the diagonal, D is just the diagonal multiplied.
swap two rows: D changes sign
take a number out of a row: it goes in front
add a multiple of a row: D stays the same
0s under the diagonal: D = the diagonal multiplied
2 Get 0s under the diagonal
D = |1, 2, 3; 0, 1, 4; 0, 1, 3|
D = |1, 2, 3; 0, 1, 4; 0, 0, −1|
Careful!
Taking away a multiple of a row changes the entries, but not D.
Careful: adding a multiple of a row: D stays the same
3 Multiply the diagonal
The diagonal: 1, 1 and −1.
Under it, all 0s.
So D is the diagonal multiplied: 1 × 1 × (−1).
D = 1 × 1 × (−1)
So D = −1.
D = −1
4 Check
Let's check with the diagonal method, on the first matrix.
D = |1, 2, 3; 0, 1, 4; 2, 5, 9|1, 2; 0, 1; 2, 5
Down: 25.
down: 1 × 1 × 9 + 2 × 4 × 2 + 3 × 0 × 5
= 25
Up: 26.
up: 2 × 1 × 3 + 5 × 4 × 1 + 9 × 0 × 2 = 26
Down minus up: D = −1.
D = 25 − 26 = −1
We get D = −1 again.
So the answer is right!
Let's look back.
First: the rules.
Then: get 0s under the diagonal.
Then: multiply the diagonal.
Last: check.
Answer
D = −1