Steply

Determinants

2 × 2 and 3 × 3 · Algebra 2 · 20 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Copy the first two columns
  2. Multiply down
  3. Multiply up
  4. Down minus up
  5. Check

The steps for Det [1, 2, 3; 0, 4, 1; 2, 1, 5]. Every solution names its steps, so you always know what comes next.

Worked examples

Det [5, 2; 3, 4] Easy

The question

Question: find the determinant of this matrix.

D = |5, 2; 3, 4|

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

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

1 Multiply the diagonals

So D = 5 × 4 − 3 × 2.

D = 5 × 4 − 3 × 2

Careful!

Always down minus up.

Up minus down gives the wrong sign.

Careful: not 6 − 20 ✗

2 Down minus up

5 times 4 is 20.

3 times 2 is 6.

D = 20 − 6

20 minus 6 is 14.

So D = 14.

D = 14

3 Check

Let's check the taking away by adding back: 14 + 6 = 20, the down product.

So D = 14 is right!

14 + 6 = 20

Let's look back.

First: multiply the diagonals.

Then: down minus up.

Last: check.

Answer
D = 14

▶ Watch it solved

Det [2, -3; 1, 4] Medium

The question

Question: find the determinant of this matrix.

D = |2, −3; 1, 4|

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

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

1 Multiply the diagonals

So D = 2 × 4 − 1 × (−3).

D = 2 × 4 − 1 × (−3)

Careful!

The up product is −3, a negative number.

Taking it away is adding: 8 − (−3) = 8 + 3.

Careful: not 8 − 3 ✗

2 Down minus up

2 times 4 is 8.

1 times (−3) is −3.

D = 8 − (−3)

Taking away a negative is adding: 8 + 3.

8 − (−3) = 8 + 3

8 minus (−3) is 11.

So D = 11.

D = 11

3 Check

Let's check the taking away by adding back: 11 + (−3) = 8, the down product.

So D = 11 is right!

11 + (−3) = 8

Let's look back.

First: multiply the diagonals.

Then: down minus up.

Last: check.

Answer
D = 11

▶ Watch it solved

Det [1, 2, 3; 0, 4, 1; 2, 1, 5] Harder

The question

Question: find the determinant of this matrix.

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

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

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

1 Copy the first two columns

We write the first two columns again, on the right.

Then every diagonal has three numbers.

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

2 Multiply down

Three diagonals go down to the right.

We multiply the three numbers on each one.

Down 1: 1 × 4 × 5.

1 × 4 × 5 is 20.

1 × 4 × 5 = 20

Down 2: 2 × 1 × 2.

2 × 1 × 2 is 4.

2 × 1 × 2 = 4

Down 3: 3 × 0 × 1.

3 × 0 × 1 is 0.

3 × 0 × 1 = 0

3 Multiply up

Three diagonals go up to the right.

We multiply those too.

Up 1: 2 × 4 × 3.

2 × 4 × 3 is 24.

2 × 4 × 3 = 24

Up 2: 1 × 1 × 1.

1 × 1 × 1 is 1.

1 × 1 × 1 = 1

Up 3: 5 × 0 × 2.

5 × 0 × 2 is 0.

5 × 0 × 2 = 0

4 Down minus up

D is all the down products, minus all the up products.

The brackets keep each group together.

D = (20 + 4 + 0) − (24 + 1 + 0)

Careful!

All three up products are taken away, not only the first.

That is why we add them up in brackets first.

Careful: not 24 − 24 + 1 + 0 ✗

The down products make 20 + 4 + 0 = 24.

The up products make 24 + 1 + 0 = 25.

D = 24 − 25

24 minus 25 is −1.

So D = −1.

D = −1

5 Check

Let's check the taking away: up plus D must give down.

25 + (−1) = 24.

So D = −1 is right!

25 + (−1) = 24

Let's look back.

First: copy the first two columns.

Then: multiply down.

Then: multiply up.

Then: down minus up.

Last: check.

Answer
D = −1

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. det [5, 2; 3, 4]
    Answer
    D = 14
  2. det [2, 1; 1, 3]
    Answer
    D = 5
  3. det [6, 3; 2, 5]
    Answer
    D = 24
  4. det [4, 2; 3, 1]
    Answer
    D = −2
  5. det [7, 2; 0, 3]
    Answer
    D = 21
  6. det [2, 4; 1, 2]
    Answer
    D = 0
  7. det [1, 3; 2, 8]
    Answer
    D = 2
  8. det [9, 4; 5, 3]
    Answer
    D = 7

Medium

  1. det [2, -3; 1, 4]
    Answer
    D = 11
  2. det [-1, 5; 2, 3]
    Answer
    D = −13
  3. det [3, -2; -4, 1]
    Answer
    D = −5
  4. det [-2, -1; 3, -4]
    Answer
    D = 11
  5. det [5, -6; 2, -1]
    Answer
    D = 7
  6. det [1, 2, 0; 3, 1, 2; 0, 1, 4]
    Answer
    D = −22
  7. det [2, 0, 1; 1, 3, 2; 1, 1, 1]
    Answer
    D = 0

Harder

  1. det [1, 2, 3; 0, 4, 1; 2, 1, 5]
    Answer
    D = −1
  2. det [2, -1, 3; 1, 0, 2; -1, 4, 1]
    Answer
    D = −1
  3. det [3, 1, -2; 0, 2, 1; 1, -1, 4]
    Answer
    D = 32
  4. det [1, 2, 3; 2, 4, 6; 1, 0, 1]
    Answer
    D = 0
  5. det [4, 0, 0; 1, 2, 0; 3, 5, 1]
    Answer
    D = 8