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

Step by step

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.

Answer