Det by row operations [0, 1, 2; 1, 2, 3; 2, 5, 1] Step by Step

Step by step

The question

Question: find the determinant by row operations of this matrix.

D = |0, 1, 2; 1, 2, 3; 2, 5, 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

D = −|1, 2, 3; 0, 1, 2; 2, 5, 1|

D = −|1, 2, 3; 0, 1, 2; 0, 1, −5|

D = −|1, 2, 3; 0, 1, 2; 0, 0, −7|

Careful!

The swap changed the sign of D.

Keep the minus in front, or the answer has the wrong sign.

Careful: a swap changes the sign of D

3 Multiply the diagonal

The diagonal: 1, 1 and −7.

Under it, all 0s.

So D is minus the diagonal multiplied: 1 × 1 × (−7).

D = −(1 × 1 × (−7))

The diagonal makes −7.

D = −(−7)

So D = 7.

D = 7

4 Check

Let's check with the diagonal method, on the first matrix.

D = |0, 1, 2; 1, 2, 3; 2, 5, 1|0, 1; 1, 2; 2, 5

Down: 16.

down: 0 × 2 × 1 + 1 × 3 × 2 + 2 × 1 × 5

= 16

Up: 9.

up: 2 × 2 × 2 + 5 × 3 × 0 + 1 × 1 × 1 = 9

Down minus up: D = 7.

D = 16 − 9 = 7

We get D = 7 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