In problems 29 through 35, let
A =

3 2 1
−5 0 2 !
; B =


4 0
1 −2
−1 7

 ; C =


2 −2 1
3 1 3
0 1 −4

 ;
D =

5 −2
1 −1
!
; E =


−3 −4 1
−3 2 1
4 0 3

 ; F =

−4 1
−3 −2
!
;
x = (−3, 5, −2); y = (4, 3)   30. a) A + B b) C + E c) 2F − 3D + I

Let's break down and solve each part of the problem systematically.

We are given matrices and vectors as:

- \( A = \begin{pmatrix} 3 & 2 & 1 \\ -5 & 0 & 2 \end{pmatrix} \)
- \( B = \begin{pmatrix} 4 & 0 \\ 1 & -2 \\ -1 & 7 \end{pmatrix} \)
- \( C = \begin{pmatrix} 2 & -2 & 1 \\ 3 & 1 & 3 \\ 0 & 1 & -4 \end{pmatrix} \)
- \( D = \begin{pmatrix} 5 & -2 \\ 1 & -1 \end{pmatrix} \)
- \( E = \begin{pmatrix} -3 & -4 & 1 \\ -3 & 2 & 1 \\ 4 & 0 & 3 \end{pmatrix} \)
- \( F = \begin{pmatrix} -4 & 1 \\ -3 & -2 \end{pmatrix} \)
- \( I \) is the identity matrix, which for a 2x2 matrix is \( I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \)

And vectors:
- \( x = (-3, 5, -2) \)
- \( y = (4, 3) \)

We need to calculate:

### 30a. \( A + B \)

Matrix addition requires both matrices to have the same dimensions. However, matrix \( A \) is a \( 2 \times 3 \) matrix, while matrix \( B \) is a \( 3 \times 2 \) matrix. Since their dimensions are different, **we cannot add these matrices**.

### 30b. \( C + E \)

Here, both \( C \) and \( E \) are \( 3 \times 3 \) matrices, so we can perform matrix addition element-wise:

\[
C + E = \begin{pmatrix} 2 & -2 & 1 \\ 3 & 1 & 3 \\ 0 & 1 & -4 \end{pmatrix} + \begin{pmatrix} -3 & -4 & 1 \\ -3 & 2 & 1 \\ 4 & 0 & 3 \end{pmatrix}
\]

Adding corresponding elements:

\[
C + E = \begin{pmatrix} 2 + (-3) & -2 + (-4) & 1 + 1 \\ 3 + (-3) & 1 + 2 & 3 + 1 \\ 0 + 4 & 1 + 0 & -4 + 3 \end{pmatrix} = \begin{pmatrix} -1 & -6 & 2 \\ 0 & 3 & 4 \\ 4 & 1 & -1 \end{pmatrix}
\]

### 30c. \( 2F - 3D + I \)

We start by calculating each term step by step:

- \( 2F \) is obtained by multiplying each element of matrix \( F \) by 2:

\[
2F = 2 \times \begin{pmatrix} -4 & 1 \\ -3 & -2 \end{pmatrix} = \begin{pmatrix} -8 & 2 \\ -6 & -4 \end{pmatrix}
\]

- \( 3D \) is obtained by multiplying each element of matrix \( D \) by 3:

\[
3D = 3 \times \begin{pmatrix} 5 & -2 \\ 1 & -1 \end{pmatrix} = \begin{pmatrix} 15 & -6 \\ 3 & -3 \end{pmatrix}
\]

- \( I \) is the identity matrix \( \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \)

Now, let's compute \( 2F - 3D \):

\[
2F - 3D = \begin{pmatrix} -8 & 2 \\ -6 & -4 \end{pmatrix} - \begin{pmatrix} 15 & -6 \\ 3 & -3 \end{pmatrix} = \begin{pmatrix} -8 - 15 & 2 - (-6) \\ -6 - 3 & -4 - (-3) \end{pmatrix} = \begin{pmatrix} -23 & 8 \\ -9 & -1 \end{pmatrix}
\]

Finally, adding the identity matrix \( I \):

\[
2F - 3D + I = \begin{pmatrix} -23 & 8 \\ -9 & -1 \end{pmatrix} + \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} -23 + 1 & 8 + 0 \\ -9 + 0 & -1 + 1 \end{pmatrix} = \begin{pmatrix} -22 & 8 \\ -9 & 0 \end{pmatrix}
\]

### Final Results:
- \( A + B \): **Cannot be added due to different dimensions.**
- \( C + E = \begin{pmatrix} -1 & -6 & 2 \\ 0 & 3 & 4 \\ 4 & 1 & -1 \end{pmatrix} \)
- \( 2F - 3D + I = \begin{pmatrix} -22 & 8 \\ -9 & 0 \end{pmatrix} \)

Do you have any questions or want further details on any part?

---

**Related Questions:**
1. What are the rules for matrix addition?
2. How do you calculate the product of a scalar and a matrix?
3. How do you perform matrix subtraction?
4. What is the purpose of an identity matrix in matrix operations?
5. How can you check if two matrices are compatible for addition or subtraction?

**Tip:** Always check the dimensions of matrices before performing operations like addition or subtraction to ensure compatibility.

In problems 29 through 35, let A = 3 2 1 −5 0 2 ; B = 4 0 1 −2 −1 7 ; C = 2 −2 1 3 1 3 0 1 −4 ; D = 5 −2 1 −1 ; E = −3 −4 1 −3 2 1 4 0 3 ; F = −4 1 −3 −2 ; x = (−3, 5, −2); y = (4, 3)   30. a) A + B b) C + E c) 2F − 3D + I

Explore how to perform matrix addition, subtraction, and scalar multiplication for matrices A, B, C, D, E, and F. This detailed solution breaks down complex matrix operations, including the addition of 3x3 matrices, scalar multiplication, and the role of the identity matrix in linear algebra. Suitable for advanced high school students in Grades 11-12.

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