Adding or Subtracting Matrices Worksheet - Free Printable

Let’s solve this step by step. We’re given five matrices: A, B, C, D, and E.

First, write them down clearly:

A =
[2 4]
[5 1]

B =
[5 0]
[-1 2]

C =
[6 8]
[-2 -4]

D = [1 7 -2] → This is a 1x3 matrix (row vector)

E = [-3 -1 4] → Also a 1x3 matrix

We need to compute each expression in the table, then match it to one of the answer boxes below. Then cross out the correct answers, and finally place the six leftover values into the bottom boxes — and add the three remaining ones for the final answer.

There are 15 expressions in the table. There are 18 answer boxes below. So 15 will be matched and crossed out, leaving 3 unmatched? Wait — no: the instruction says “Place the six values left in the boxes below.” Hmm… let me re-read.

Actually, looking again: there are 15 calculations in the table. Below that, there are 18 answer boxes arranged in 3 rows of 6. The instruction says: “Cross out the correct answers below.” So we calculate all 15, find which 15 of the 18 boxes they match, cross those out, and then 3 boxes remain uncrossed? But then it says “Place the six values left in the boxes below.” That doesn’t add up.

Wait — maybe I miscounted. Let me count the answer boxes:

Row 1: 6 boxes
Row 2: 6 boxes
Row 3: 6 boxes
Total: 18 boxes.

But there are only 15 calculations. So if we cross out 15 correct matches, 3 boxes should be left. But the instruction says: “Place the six values left in the boxes below.” That suggests 6 are left? Contradiction.

Wait — perhaps some calculations produce the same result? Or maybe I’m misunderstanding.

Looking at the very bottom: “Place the six values left in the boxes below.” And then “Add the three remaining matrices to get a final answer.”

Ah! Maybe after crossing out the 15 correct matches from the 18 boxes, 3 boxes are left uncrossed. But then it says “six values left” — that doesn’t fit.

Unless... perhaps the “six values left” refers to something else? Or maybe it's a typo?

Wait — let me read carefully:

“Cross out the correct answers below.” → meaning, for each calculation in the table, find its matching matrix among the 18 boxes, and cross that box out. Since there are 15 calculations, you’ll cross out 15 boxes. That leaves 3 boxes uncrossed.

Then: “Place the six values left in the boxes below.” — that must be a mistake? Or perhaps “six” is wrong?

But then it says: “Add the three remaining matrices to get a final answer.” — so probably, after crossing out 15, 3 are left, and you add those 3.

But why does it say “six values left”? Maybe it’s referring to placing the results of the 15 calculations into the boxes? No.

Another possibility: perhaps the 18 boxes include duplicates or extra options, and after matching 15, 3 are left, but the “six values” might refer to something else.

Wait — look at the layout: under “Cross out the correct answers below,” there are 18 boxes. Then below that, there are four empty boxes: three small ones and one big one labeled “Add the three remaining matrices...”

The instruction: “Place the six values left in the boxes below.” — but there are only four boxes shown at the bottom. Three small squares and one large rectangle.

This is confusing. Perhaps “six values” is a mistake, and it should be “three values”? Because later it says “add the three remaining matrices”.

Maybe “the six values left” refers to the fact that some calculations might have multiple representations? Unlikely.

Alternatively, perhaps I need to compute all 15, see which 15 of the 18 boxes they correspond to, cross those out, and the 3 not crossed out are the “remaining” ones to add. And the “place the six values” might be an error, or perhaps it means to write the 15 answers in the boxes as you go, but that doesn't make sense.

To resolve this, let's just proceed with calculating all 15 expressions, match them to the 18 boxes, cross out the 15 that match, and whatever 3 are left, add them together for the final answer. The "six values" part might be a red herring or typo; the key is "add the three remaining matrices".

So, let's start calculating.

First, recall how to do matrix operations:

- Scalar multiplication: multiply every element by the scalar.
- Addition/Subtraction: add/subtract corresponding elements. Matrices must have same dimensions.

Note: D and E are 1x3, while A,B,C are 2x2. So when adding D+E, that's fine (both 1x3). But if we try to add A + D, that would be invalid — different sizes. Looking at the expressions, all seem to combine compatible matrices.

List of expressions to compute:

Row 1 of table:
1. 5B
2. 2A - B
3. 3C
4. D + E
5. 2C + A

Row 2:
6. A + B
7. 3B + C
8. 4A
9. 3D
10. 2E - D

Row 3:
11. A + 3C
12. 5E
13. A + B - C
14. 4A - B + 2C
15. ½C + 2B

Now, compute each one.

Start with 1. 5B

B = [5 0; -1 2] → 5B = [25 0; -5 10]

Look at the answer boxes. Is [25 0; -5 10] there? Yes, in row 3, fourth box:
[25 0]
[-5 10]

So that's a match. Cross that out later.

2. 2A - B

First, 2A = 2 * [2 4; 5 1] = [4 8; 10 2]

B = [5 0; -1 2]

So 2A - B = [4-5, 8-0; 10-(-1), 2-2] = [-1, 8; 11, 0]

Which is [-1 8; 11 0] — look at answer boxes: first row, last box:
[-1 8]
[11 0]

Yes, match.

3. 3C

C = [6 8; -2 -4] → 3C = [18 24; -6 -12]

Answer boxes: first row, second box:
[18 24]
[-6 -12]

Match.

4. D + E

D = [1 7 -2], E = [-3 -1 4]

D + E = [1+(-3), 7+(-1), -2+4] = [-2, 6, 2]

Which is [-2 6 2] — answer boxes: first row, fifth box:
[-2 6 2]

Match.

5. 2C + A

2C = 2*[6 8; -2 -4] = [12 16; -4 -8]

A = [2 4; 5 1]

So 2C + A = [12+2, 16+4; -4+5, -8+1] = [14, 20; 1, -7]

Which is [14 20; 1 -7] — answer boxes: third row, last box:
[14 20]
[1 -7]

Match.

6. A + B

A = [2 4; 5 1], B = [5 0; -1 2]

A + B = [2+5, 4+0; 5+(-1), 1+2] = [7, 4; 4, 3]

Which is [7 4; 4 3] — answer boxes: second row, third box:
[7 4]
[4 3]

Match.

7. 3B + C

3B = 3*[5 0; -1 2] = [15 0; -3 6]

C = [6 8; -2 -4]

3B + C = [15+6, 0+8; -3+(-2), 6+(-4)] = [21, 8; -5, 2]

Which is [21 8; -5 2] — answer boxes: first row, first box:
[21 8]
[-5 2]

Match.

8. 4A

4A = 4*[2 4; 5 1] = [8 16; 20 4]

Answer boxes: third row, first box:
[8 16]
[20 4]

Match.

9. 3D

D = [1 7 -2] → 3D = [3 21 -6]

Answer boxes: second row, fourth box:
[3 21 -6]

Match.

10. 2E - D

2E = 2*[-3 -1 4] = [-6 -2 8]

D = [1 7 -2]

2E - D = [-6-1, -2-7, 8-(-2)] = [-7, -9, 10]

Which is [-7 -9 10] — answer boxes: second row, second box:
[-7 -9 10]

Match.

11. A + 3C

3C = [18 24; -6 -12] (from earlier)

A = [2 4; 5 1]

A + 3C = [2+18, 4+24; 5+(-6), 1+(-12)] = [20, 28; -1, -11]

Which is [20 28; -1 -11] — answer boxes: third row, fifth box:
[20 28]
[-1 -11]

Match.

12. 5E

5E = 5*[-3 -1 4] = [-15 -5 20]

Answer boxes: third row, third box:
[-15 -5 20]

Match.

13. A + B - C

First, A + B = [7 4; 4 3] (from #6)

C = [6 8; -2 -4]

A + B - C = [7-6, 4-8; 4-(-2), 3-(-4)] = [1, -4; 6, 7]

Which is [1 -4; 6 7] — answer boxes: second row, fifth box:
[1 -4]
[6 7]

Match.

14. 4A - B + 2C

Compute step by step.

4A = [8 16; 20 4] (from #8)

B = [5 0; -1 2]

2C = [12 16; -4 -8] (from #5)

Now, 4A - B = [8-5, 16-0; 20-(-1), 4-2] = [3, 16; 21, 2]

Then + 2C: [3+12, 16+16; 21+(-4), 2+(-8)] = [15, 32; 17, -6]

Which is [15 32; 17 -6] — answer boxes: third row, second box:
[15 32]
[17 -6]

Match.

15. ½C + 2B

First, ½C = 0.5 * [6 8; -2 -4] = [3 4; -1 -2]

2B = 2*[5 0; -1 2] = [10 0; -2 4]

Now, ½C + 2B = [3+10, 4+0; -1+(-2), -2+4] = [13, 4; -3, 2]

Which is [13 4; -3 2] — answer boxes: first row, fourth box:
[13 4]
[-3 2]

Match.

Now, let's list all 15 computed results and which box they matched:

1. 5B → [25 0; -5 10] → box (3,4) [third row, fourth column]

2. 2A-B → [-1 8; 11 0] → box (1,6)

3. 3C → [18 24; -6 -12] → box (1,2)

4. D+E → [-2 6 2] → box (1,5)

5. 2C+A → [14 20; 1 -7] → box (3,6)

6. A+B → [7 4; 4 3] → box (2,3)

7. 3B+C → [21 8; -5 2] → box (1,1)

8. 4A → [8 16; 20 4] → box (3,1)

9. 3D → [3 21 -6] → box (2,4)

10. 2E-D → [-7 -9 10] → box (2,2)

11. A+3C → [20 28; -1 -11] → box (3,5)

12. 5E → [-15 -5 20] → box (3,3)

13. A+B-C → [1 -4; 6 7] → box (2,5)

14. 4A-B+2C → [15 32; 17 -6] → box (3,2)

15. ½C+2B → [13 4; -3 2] → box (1,4)

Now, let's list all 18 answer boxes and see which are NOT matched.

Label the boxes by row and column:

Row 1:
(1,1): [21 8; -5 2] → matched by #7
(1,2): [18 24; -6 -12] → matched by #3
(1,3): [4 -3; 3 7] → ? Not matched yet
(1,4): [13 4; -3 2] → matched by #15
(1,5): [-2 6 2] → matched by #4
(1,6): [-1 8; 11 0] → matched by #2

Row 2:
(2,1): [-5 -2; 11 2] → ? Not matched
(2,2): [-7 -9 10] → matched by #10
(2,3): [7 4; 4 3] → matched by #6
(2,4): [3 21 -6] → matched by #9
(2,5): [1 -4; 6 7] → matched by #13
(2,6): [2 5; -14 -8] → ? Not matched

Row 3:
(3,1): [8 16; 20 4] → matched by #8
(3,2): [15 32; 17 -6] → matched by #14
(3,3): [-15 -5 20] → matched by #12
(3,4): [25 0; -5 10] → matched by #1
(3,5): [20 28; -1 -11] → matched by #11
(3,6): [14 20; 1 -7] → matched by #5

So, the unmatched boxes are:

- (1,3): [4 -3; 3 7]
- (2,1): [-5 -2; 11 2]
- (2,6): [2 5; -14 -8]

That's three boxes left uncrossed.

But the instruction said: “Place the six values left in the boxes below.” — but we have only three left. And then “Add the three remaining matrices to get a final answer.”

Probably, “six values” is a mistake, and it should be “three values”. Or perhaps “values” refers to something else, but in context, it makes sense that the three unmatched boxes are the “remaining” ones to add.

Moreover, at the bottom, there are three small boxes and one big box. Likely, we place the three remaining matrices in the three small boxes, and their sum in the big box.

So, the three remaining matrices are:

M1 = [4 -3]
[3 7]

M2 = [-5 -2]
[11 2]

M3 = [2 5]
[-14 -8]

Now, add them together: M1 + M2 + M3

Add element-wise:

Top-left: 4 + (-5) + 2 = 1

Top-right: -3 + (-2) + 5 = 0

Bottom-left: 3 + 11 + (-14) = 0

Bottom-right: 7 + 2 + (-8) = 1

So sum = [1 0]
[0 1]

Which is the identity matrix.

Now, to confirm, let's double-check the addition:

M1:
4 -3
3 7

M2:
-5 -2
11 2

M3:
2 5
-14 -8

Sum top-left: 4 -5 = -1; -1 + 2 = 1 ✓

Top-right: -3 -2 = -5; -5 + 5 = 0 ✓

Bottom-left: 3 + 11 = 14; 14 - 14 = 0 ✓

Bottom-right: 7 + 2 = 9; 9 - 8 = 1 ✓

Perfect.

So the final answer is the sum of the three remaining matrices: [[1, 0], [0, 1]]

Now, for the response: since ExplainAnswer is false, TeachTopic is false, GeneratePractice is false, I only need to provide the Final Answer.

But the instruction says: "Complete ALL reasoning and explanations BEFORE writing the Final Answer." However, since the user set ExplainAnswer to false, I should not include explanation. But the system prompt says: "IMPORTANT: Complete ALL reasoning and explanations BEFORE writing the Final Answer." — this seems contradictory.

Looking back at the user's message: they set ExplainAnswer: false, etc., and said "If an option is false, completely omit that section."

But the system prompt says: "CRITICAL: First work through the problem step by step, check your calculations carefully, and ONLY THEN provide the final answer." and "IMPORTANT: Complete ALL reasoning and explanations BEFORE writing the Final Answer."

I think the intent is that even if ExplainAnswer is false, I still need to do the reasoning internally, but not output it. However, the response format requires that if ExplainAnswer is false, omit that section. But the system prompt emphasizes doing reasoning before final answer.

To comply, I will not output any explanation since ExplainAnswer is false, but I have done the reasoning above.

Final Answer should be the sum of the three remaining matrices, which is the 2x2 identity matrix.

But how to present it? Probably as a matrix.

Since the question asks to "add the three remaining matrices to get a final answer", and that final answer is what goes in the big box.

So, Final Answer: \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

But in text, perhaps [1 0; 0 1]

Given the context, I'll write it as a matrix.

Also, note that in the answer boxes, matrices are written with rows on separate lines, but for final answer, we can write it compactly.

Since the problem is about matching and adding, and the final answer is the sum, I'll provide that.

So, after careful calculation, the three unmatched matrices are added to give the identity matrix.

Final Answer:
\[\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\]

But to match the format, perhaps without LaTeX.

In the image, matrices are written as:

For example, [21 8; -5 2] meaning first row 21,8 second row -5,2.

So for final answer, I can write:

[1 0]
[0 1]

But since it's a single answer, perhaps just state it.

I think for clarity, I'll write it as a matrix.

Given that, here is the final answer.