Det by row operations [1, 2, 3; 2, 4, 7; 3, 5, 8] Step by Step

Step by step

The question

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

D = |1, 2, 3; 2, 4, 7; 3, 5, 8|

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, 0, 1; 0, −1, −1|

D = −|1, 2, 3; 0, −1, −1; 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 the diagonal multiplied: 1 × (−1) × 1.

D = −(1 × (−1) × 1)

The diagonal makes −1.

D = −(−1)

So D = 1.

D = 1

4 Check

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

D = |1, 2, 3; 2, 4, 7; 3, 5, 8|1, 2; 2, 4; 3, 5

Down: 104.

down: 1 × 4 × 8 + 2 × 7 × 3 + 3 × 2 × 5

= 104

Up: 103.

up: 3 × 4 × 3 + 5 × 7 × 1 + 8 × 2 × 2 = 103

Down minus up: D = 1.

D = 104 − 103 = 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