Math worksheet with matrix operations problems, including addition, subtraction, and scalar multiplication, with instructions to simplify and write "undefined" for undefined expressions.

Let’s work through each problem one by one. We’ll check if the operations are defined, and if so, compute them step by step.

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Problem 1:

Multiply two 2x2 matrices:

Matrix A =
[ 2 -1 ]
[-6 1 ]

Matrix B =
[ 4 4 ]
[-3 -5 ]

To multiply, we do row × column.

First row, first column: (2)(4) + (-1)(-3) = 8 + 3 = 11
First row, second column: (2)(4) + (-1)(-5) = 8 + 5 = 13
Second row, first column: (-6)(4) + (1)(-3) = -24 - 3 = -27
Second row, second column: (-6)(4) + (1)(-5) = -24 - 5 = -29

Result:
[11 13]
[-27 -29]

✔ Defined and computed.

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Problem 2:

Add three 2x2 matrices:

A = [2 6] B = [5 3] C = [1 2]
[-6 4] [-6 2] [2 0]

Add element by element:

Top left: 2 + 5 + 1 = 8
Top right: 6 + 3 + 2 = 11
Bottom left: -6 + (-6) + 2 = -10
Bottom right: 4 + 2 + 0 = 6

Result:
[8 11]
[-10 6]

✔ Defined and computed.

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Problem 3:

Multiply a 2x2 matrix by a 2x2 matrix:

A = [-1 5] B = [-3 6]
[ 5 -5] [-3 0]

Check dimensions: 2x2 × 2x2 → OK.

Compute:

Row 1, Col 1: (-1)(-3) + (5)(-3) = 3 - 15 = -12
Row 1, Col 2: (-1)(6) + (5)(0) = -6 + 0 = -6
Row 2, Col 1: (5)(-3) + (-5)(-3) = -15 + 15 = 0
Row 2, Col 2: (5)(6) + (-5)(0) = 30 + 0 = 30

Result:
[-12 -6]
[ 0 30]

✔ Defined and computed.

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Problem 4:

Add a 2x2 matrix and a 2x1 matrix:

A = [1 -6] B = [-3]
[3 5] [ 0]
[ 3]
[-2]

Wait — Matrix A is 2x2, Matrix B is 4x1. You can only add matrices of the same size.

These are NOT the same size → undefined

✘ Undefined.

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Problem 5:

Add two 4x1 matrices (column vectors):

A = [-2] B = [-4]
[-3] [ 6]
[-6] [ 0]
[ 2] [-3]

Add element by element:

-2 + (-4) = -6
-3 + 6 = 3
-6 + 0 = -6
2 + (-3) = -1

Result:
[-6]
[ 3]
[-6]
[-1]

✔ Defined and computed.

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Problem 6:

Scalar multiplication and subtraction:

First, compute scalar multiples:

-4 × [ -3 -6 ] = [ 12 24 ]
[ 1 4 ] [ -4 -16 ]

-2 × [ -2 6 ] = [ 4 -12 ]
[ -1 -4 ] [ 2 8 ]

Now subtract: [12 24] [ 4 -12] = [12-4 24-(-12)] = [8 36]
[-4 -16] - [ 2 8] [-4-2 -16-8] [-6 -24]

Result:
[ 8 36]
[-6 -24]

✔ Defined and computed.

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Problem 7:

Three matrices:

A = [3 1 3] B = [-1 3] C = [0 -6 -1]
[0 5 -3] [6 1] [1 1 4]

Dimensions:

A: 2x3
B: 2x2
C: 2x3

We are to compute: A + B + C

But B is 2x2, while A and C are 2x3 → cannot add matrices of different sizes.

So even though A and C match, B does not → undefined

✘ Undefined.

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Problem 8:

Scalar multiplication and addition:

First, -5 × [ 2 ] = [ -10 ]
[ 2 ] [ -10 ]
[-6] [ 30 ]

Then add [ 2 ]
[-4]
[ 4 ]

Element-wise:

-10 + 2 = -8
-10 + (-4) = -14
30 + 4 = 34

Result:
[-8]
[-14]
[34]

✔ Defined and computed.

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Final Answers:

1) [11 13]
[-27 -29]

2) [8 11]
[-10 6]

3) [-12 -6]
[ 0 30]

4) undefined

5) [-6]
[ 3]
[-6]
[-1]

6) [ 8 36]
[-6 -24]

7) undefined

8) [-8]
[-14]
[34]

Final Answer:
1) $\begin{bmatrix} 11 & 13 \\ -27 & -29 \end{bmatrix}$
2) $\begin{bmatrix} 8 & 11 \\ -10 & 6 \end{bmatrix}$
3) $\begin{bmatrix} -12 & -6 \\ 0 & 30 \end{bmatrix}$
4) undefined
5) $\begin{bmatrix} -6 \\ 3 \\ -6 \\ -1 \end{bmatrix}$
6) $\begin{bmatrix} 8 & 36 \\ -6 & -24 \end{bmatrix}$
7) undefined
8) $\begin{bmatrix} -8 \\ -14 \\ 34 \end{bmatrix}$