Det by row operations [0, 2, 4; 1, 1, 1; 2, 3, 5] Step by Step

Step by step

The question

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

D = |0, 2, 4; 1, 1, 1; 2, 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, 1, 1; 0, 2, 4; 2, 3, 5|

D = −|1, 1, 1; 0, 2, 4; 0, 1, 3|

D = −2 × |1, 1, 1; 0, 1, 2; 0, 1, 3|

D = −2 × |1, 1, 1; 0, 1, 2; 0, 0, 1|

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 1.

Under it, all 0s.

So D is minus 2 times the diagonal multiplied: 1 × 1 × 1.

D = −2 × (1 × 1 × 1)

The diagonal makes 1.

D = −2 × 1

So D = −2.

D = −2

4 Check

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

D = |0, 2, 4; 1, 1, 1; 2, 3, 5|0, 2; 1, 1; 2, 3

Down: 16.

down: 0 × 1 × 5 + 2 × 1 × 2 + 4 × 1 × 3

= 16

Up: 18.

up: 2 × 1 × 4 + 3 × 1 × 0 + 5 × 1 × 2 = 18

Down minus up: D = −2.

D = 16 − 18 = −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