Steply

Matrices

Size, one entry, + −, times a number · Algebra 2 · 22 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Check the sizes
  2. Times first
  3. Add the matching entries
  4. Check

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

Worked examples

Size of [2, -1, 0; 5, 3, 7] Easy

The question

Question: find the size of this matrix.

[2, −1, 0; 5, 3, 7]

A matrix is numbers in rows and columns, inside big brackets.

Each number in a matrix is called an entry.

1 Count the rows

Rows go across.

We count them: 1, 2.

There are 2 rows.

rows = 2

2 Count the columns

Columns go down.

We count them: 1, 2, 3.

There are 3 columns.

columns = 3

The size is rows by columns: 2 by 3.

Rows come first.

We write it like this, and say 2 by 3.

size = 2 × 3

Careful!

Rows always come first, then columns.

3 by 2 would be a different shape.

Careful: not 3 × 2 ✗

3 Check

Let's check.

2 rows of 3 make 2 × 3 = 6 entries.

We count all the entries in the matrix: 6.

So the size is right!

2 × 3 = 6

Let's look back.

First: count the rows.

Then: count the columns.

Last: check.

Answer
Careful: not 3 × 2 ✗

▶ Watch it solved

Calculate [-2, 5; 3, -1] + [4, -3; -6, 2] Medium

The question

Question: work out this sum of two matrices.

[−2, 5; 3, −1] + [4, −3; −6, 2]

A matrix is numbers in rows and columns, inside big brackets.

Each number in a matrix is called an entry.

1 Check the sizes

Both are 2 by 2: the same size.

Good!

We can only add matrices of the same size, so every entry has a partner in the same place.

2 Add the matching entries

We add the numbers in the same place.

Top left: −2 and 4.

So we write −2 + 4 there.

And the same in every place.

= [−2 + 4, 5 + (−3); 3 + (−6), −1 + 2]

Careful!

We add only the two entries in the same place: top left with top left, and so on.

Careful: only entries in the same place

Now we work out each one.

The answer: rows 2, 2 and −3, 1.

= [2, 2; −3, 1]

3 Check

Let's check by taking away.

The answer minus the second matrix must give back the first one.

2 − 4 = −2, 2 − (−3) = 5
−3 − (−6) = 3, 1 − 2 = −1

We get the first matrix back.

So the answer is right!

Let's look back.

First: check the sizes.

Then: add the matching entries.

Last: check.

Answer
= [2, 2; −3, 1]

▶ Watch it solved

Calculate 2[1, 3; -1, 0] + [4, -2; 5, 1] Harder

The question

Question: work out this.

2[1, 3; −1, 0] + [4, −2; 5, 1]

A matrix is numbers in rows and columns, inside big brackets.

Each number in a matrix is called an entry.

1 Check the sizes

Both are 2 by 2: the same size.

Good!

We can only add matrices of the same size, so every entry has a partner in the same place.

2 Times first

Like the order of operations: times comes before plus and minus.

2 in front means: every entry times 2.

We write 2 × in front of each one.

= [2 × 1, 2 × 3; 2 × (−1), 2 × 0] + [4, −2; 5, 1]

Now we work out each one.

2 times 1 is 2, and so on.

= [2, 6; −2, 0] + [4, −2; 5, 1]

3 Add the matching entries

We add the numbers in the same place.

Top left: 2 and 4.

So we write 2 + 4 there.

And the same in every place.

= [2 + 4, 6 + (−2); −2 + 5, 0 + 1]

Careful!

We add only the two entries in the same place: top left with top left, and so on.

Careful: only entries in the same place

Now we work out each one.

The answer: rows 6, 4 and 3, 1.

= [6, 4; 3, 1]

4 Check

Let's check by taking away.

The answer minus the second matrix must give back the first one.

6 − 4 = 2, 4 − (−2) = 6
3 − 5 = −2, 1 − 1 = 0

We get the first matrix back, after the times.

So the answer is right!

Let's look back.

First: check the sizes.

Then: times first.

Then: add the matching entries.

Last: check.

Answer
= [6, 4; 3, 1]

▶ Watch it solved

Practice

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

Easy

  1. size of [2, -1, 0; 5, 3, 7]
    Answer
    Careful: not 3 × 2 ✗
  2. size of [4; -2; 9]
    Answer
    Careful: not 1 × 3 ✗
  3. size of [3, 8]
    Answer
    Careful: not 2 × 1 ✗
  4. find a_21 in [5, 1; -3, 7]
    Answer
    Careful: not a12 = 1 ✗
  5. find a_13 in [2, 0, 6; 1, 4, -5]
    Answer
    Careful: first the row, then the column
  6. [2, 1; 0, 3] + [4, 5; 1, 2]
    Answer
    = [6, 6; 1, 5]
  7. [7, 4; 6, 9] - [3, 1; 2, 5]
    Answer
    = [4, 3; 4, 4]
  8. 3[2, 1; 0, 4]
    Answer
    = [6, 3; 0, 12]
  9. [1, 3, 2] + [4, 0, 5]
    Answer
    = [5, 3, 7]

Medium

  1. [-2, 5; 3, -1] + [4, -3; -6, 2]
    Answer
    = [2, 2; −3, 1]
  2. [6, 0; -1, 4] - [2, -3; 5, 1]
    Answer
    = [4, 3; −6, 3]
  3. -2[3, -1; 0, 5]
    Answer
    = [−6, 2; 0, −10]
  4. 4[1, -2, 3]
    Answer
    = [4, −8, 12]
  5. [5, -2, 1; 0, 3, -4] + [-1, 2, 6; 3, -3, 2]
    Answer
    = [4, 0, 7; 3, 0, −2]
  6. find a_32 in [1, 4; -2, 6; 9, 0]
    Answer
    Careful: first the row, then the column
  7. [1, 2; 3, 4; 5, 6] - [6, 5; 4, 3; 2, 1]
    Answer
    = [−5, −3; −1, 1; 3, 5]
  8. size of [0, 1, -1, 2; 3, 5, 8, 4]
    Answer
    Careful: not 4 × 2 ✗
  9. 5[2; -1; 3]
    Answer
    = [10; −5; 15]

Harder

  1. 2[1, 3; -1, 0] + [4, -2; 5, 1]
    Answer
    = [6, 4; 3, 1]
  2. 3[2, -1; 4, 0] - 2[1, 2; -3, 1]
    Answer
    = [4, −7; 18, −2]
  3. [4, 2; 1, 3] - 3[1, 0; 2, -1]
    Answer
    = [1, 2; −5, 6]
  4. [1, 2] + [3; 4]
    Answer
    1 × 2 and 2 × 1: not the same size