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

Step by step

The question

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

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

D = 2 × |1, 2, 1; 0, 1, 3; 0, 1, 5|

D = 2 × |1, 2, 1; 0, 1, 3; 0, 0, 2|

Careful!

The 2 we took out stays in front.

D is 2 times the new determinant.

Careful: the 2 stays in front

3 Multiply the diagonal

The diagonal: 1, 1 and 2.

Under it, all 0s.

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

D = 2 × (1 × 1 × 2)

The diagonal makes 2.

D = 2 × 2

So D = 4.

D = 4

4 Check

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

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

Down: 32.

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

= 32

Up: 28.

up: 0 × 3 × 2 + 1 × 4 × 2 + 5 × 1 × 4 = 28

Down minus up: D = 4.

D = 32 − 28 = 4

We get D = 4 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