What is the determinant of the matrix [3 2; 5 4]? Step by Step

Step by step

The question

Question: find the determinant of this matrix.

D = |3, 2; 5, 4|

The determinant is one number we get from a square matrix.

We write the matrix between two straight lines, and call it D.

Later, D helps us solve equations with matrices.

There are four choices.

First we solve it, the normal way.

Then we pick.

1 Multiply the diagonals

Down the first diagonal: 3 × 4.

Up the other diagonal: 5 × 2.

The determinant is down minus up.

So D = 3 × 4 − 5 × 2.

D = 3 × 4 − 5 × 2

Careful!

Always down minus up.

Up minus down gives the wrong sign.

Careful: not 5 × 2 − 3 × 4 ✗

2 Down minus up

3 times 4 is 12.

5 times 2 is 10.

D = 12 − 10

12 minus 10 is 2.

So D = 2.

D = 2

3 Check

Let's check the taking away by adding back: 2 + 10 = 12, the first product.

So D = 2 is right!

2 + 10 = 12

4 Pick the choice

Our answer is D = 2.

That is choice B.

B) 2

5 Why not the others

A is −2.

Wrong: 10 − 12: took the products the wrong way round.

That is the Careful mistake from before.

A) −2 ✗ 10 − 12: took the products the wrong way round

C is 7.

Wrong: 3 + 4: added the main diagonal; we multiply.

C) 7 ✗ 3 + 4: added the main diagonal; we multiply

D is 22.

Wrong: 12 + 10: added the products; we take away.

D) 22 ✗ 12 + 10: added the products; we take away

Let's look back.

First: multiply the diagonals.

Then: down minus up.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer