Worked examples
The question
Question: find the determinant by row operations of this matrix.
D = |2, 1, 5; 0, 3, 4; 0, 0, −1|
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
There are already 0s under the diagonal.
No moves are needed.
Careful!
Only the diagonal is multiplied.
The numbers above it do not matter.
Careful: multiply only the diagonal
3 Multiply the diagonal
The diagonal: 2, 3 and −1.
Under it, all 0s.
So D is the diagonal multiplied: 2 × 3 × (−1).
D = 2 × 3 × (−1)
So D = −6.
D = −6
4 Check
Let's check with the diagonal method, on the first matrix.
D = |2, 1, 5; 0, 3, 4; 0, 0, −1|2, 1; 0, 3; 0, 0
Down: −6.
down: 2 × 3 × (−1) + 1 × 4 × 0 + 5 × 0 × 0
= −6
Up: 0.
up: 0 × 3 × 5 + 0 × 4 × 2 + (−1) × 0 × 1 = 0
Down minus up: D = −6.
D = −6 − 0 = −6
We get D = −6 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.
▶ Watch it solved
The question
Question: find the determinant by row operations of this matrix.
D = |1, 2, 3; 2, 5, 7; 1, 3, 5|
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, 1; 0, 1, 2|
D = |1, 2, 3; 0, 1, 1; 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; 2, 5, 7; 1, 3, 5|1, 2; 2, 5; 1, 3
Down: 57.
down: 1 × 5 × 5 + 2 × 7 × 1 + 3 × 2 × 3
= 57
Up: 56.
up: 1 × 5 × 3 + 3 × 7 × 1 + 5 × 2 × 2 = 56
Down minus up: D = 1.
D = 57 − 56 = 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.
▶ Watch it solved
The question
Question: find the determinant by row operations of this matrix.
D = |3, 6, 3; 1, 4, 2; 2, 6, 7|
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 = 3 × |1, 2, 1; 1, 4, 2; 2, 6, 7|
D = 3 × |1, 2, 1; 0, 2, 1; 0, 2, 5|
D = 3 × |1, 2, 1; 0, 2, 1; 0, 0, 4|
Careful!
The 3 we took out stays in front.
D is 3 times the new determinant.
Careful: the 3 stays in front
3 Multiply the diagonal
The diagonal: 1, 2 and 4.
Under it, all 0s.
So D is 3 times the diagonal multiplied: 1 × 2 × 4.
D = 3 × (1 × 2 × 4)
The diagonal makes 8.
D = 3 × 8
So D = 24.
D = 24
4 Check
Let's check with the diagonal method, on the first matrix.
D = |3, 6, 3; 1, 4, 2; 2, 6, 7|3, 6; 1, 4; 2, 6
Down: 126.
down: 3 × 4 × 7 + 6 × 2 × 2 + 3 × 1 × 6
= 126
Up: 102.
up: 2 × 4 × 3 + 6 × 2 × 3 + 7 × 1 × 6 = 102
Down minus up: D = 24.
D = 126 − 102 = 24
We get D = 24 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.
▶ Watch it solved
