Step by step
The question
Question: find the determinant by row operations of this matrix.
D = |1, 2; 3, 4|
We call the rows R 1 and R 2.
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; 0, −2|
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 and −2.
Under it, all 0s.
So D is the diagonal multiplied: 1 × (−2).
D = 1 × (−2)
So D = −2.
D = −2
4 Check
Let's check with down minus up.
D = |1, 2; 3, 4|
So D = 1 × 4 − 3 × 2.
D = 1 × 4 − 3 × 2
1 times 4 is 4.
3 times 2 is 6.
D = 4 − 6
4 minus 6 is −2.
So D = −2.
D = −2
We get D = −2 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 = −2