Glven the following matrices, perform the following: Glven the following: $$ A=\left[\begin{array}{ccc} 3 & 2 & -1 \ 7 & 3 & -2 \ 6 & 1 & 0 \end{array}\right] $$ $$ B=\left[\begin{array}{ccc} -1 & 4 & 2 \ 6 & 6 & -1 \ -5 & 2 & 6 \end{array}\right] $$ Find $$ A+B $$ Find $$ A-B $$ Find $$ 2 A $$ Find 3B Find $$ \quad 2 A-3 B $$ Find $$ \quad 3 B-2 A $$ Additive Inverse Matrix (for $$ 3 \times 3 $$ ) Find $$ -A $$

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$$ A = \begin{bmatrix} 3 & 2 & -1 \ 7 & 3 & -2 \ 6 & 1 & 0 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} -1 & 4 & 2 \ 6 & 6 & -1 \ -5 & 2 & 6 \end{bmatrix} $$

**1. Calculate $$ A + B $$:**

Add corresponding elements:

- First row: 
  - $$ 3 + (-1) = 2 $$
  - $$ 2 + 4 = 6 $$
  - $$ -1 + 2 = 1 $$

- Second row:
  - $$ 7 + 6 = 13 $$
  - $$ 3 + 6 = 9 $$
  - $$ -2 + (-1) = -3 $$

- Third row:
  - $$ 6 + (-5) = 1 $$
  - $$ 1 + 2 = 3 $$
  - $$ 0 + 6 = 6 $$

Thus,
$$
A + B = \begin{bmatrix} 2 & 6 & 1 \ 13 & 9 & -3 \ 1 & 3 & 6 \end{bmatrix}
$$

**2. Calculate $$ A - B $$:**

Subtract corresponding elements:

- First row: 
  - $$ 3 - (-1) = 4 $$
  - $$ 2 - 4 = -2 $$
  - $$ -1 - 2 = -3 $$

- Second row:
  - $$ 7 - 6 = 1 $$
  - $$ 3 - 6 = -3 $$
  - $$ -2 - (-1) = -1 $$

- Third row:
  - $$ 6 - (-5) = 11 $$
  - $$ 1 - 2 = -1 $$
  - $$ 0 - 6 = -6 $$

Thus,
$$
A - B = \begin{bmatrix} 4 & -2 & -3 \ 1 & -3 & -1 \ 11 & -1 & -6 \end{bmatrix}
$$

**3. Calculate $$ 2A $$:**

Multiply every element in $$ A $$ by 2:

- First row: 
  - $$ 2 \times 3 = 6 $$
  - $$ 2 \times 2 = 4 $$
  - $$ 2 \times (-1) = -2 $$

- Second row:
  - $$ 2 \times 7 = 14 $$
  - $$ 2 \times 3 = 6 $$
  - $$ 2 \times (-2) = -4 $$

- Third row:
  - $$ 2 \times 6 = 12 $$
  - $$ 2 \times 1 = 2 $$
  - $$ 2 \times 0 = 0 $$

Thus,
$$
2A = \begin{bmatrix} 6 & 4 & -2 \ 14 & 6 & -4 \ 12 & 2 & 0 \end{bmatrix}
$$

**4. Calculate $$ 3B $$:**

Multiply every element in $$ B $$ by 3:

- First row:
  - $$ 3 \times (-1) = -3 $$
  - $$ 3 \times 4 = 12 $$
  - $$ 3 \times 2 = 6 $$

- Second row:
  - $$ 3 \times 6 = 18 $$
  - $$ 3 \times 6 = 18 $$
  - $$ 3 \times (-1) = -3 $$

- Third row:
  - $$ 3 \times (-5) = -15 $$
  - $$ 3 \times 2 = 6 $$
  - $$ 3 \times 6 = 18 $$

Thus,
$$
3B = \begin{bmatrix} -3 & 12 & 6 \ 18 & 18 & -3 \ -15 & 6 & 18 \end{bmatrix}
$$

**5. Calculate $$ 2A - 3B $$:**

Subtract corresponding elements from $$ 2A $$ and $$ 3B $$:

- First row:
  - $$ 6 - (-3) = 9 $$
  - $$ 4 - 12 = -8 $$
  - $$ -2 - 6 = -8 $$

- Second row:
  - $$ 14 - 18 = -4 $$
  - $$ 6 - 18 = -12 $$
  - $$ -4 - (-3) = -1 $$

- Third row:
  - $$ 12 - (-15) = 27 $$
  - $$ 2 - 6 = -4 $$
  - $$ 0 - 18 = -18 $$

Thus,
$$
2A - 3B = \begin{bmatrix} 9 & -8 & -8 \ -4 & -12 & -1 \ 27 & -4 & -18 \end{bmatrix}
$$

**6. Calculate $$ 3B - 2A $$:**

Subtract corresponding elements from $$ 3B $$ and $$ 2A $$:

- First row:
  - $$ -3 - 6 = -9 $$
  - $$ 12 - 4 = 8 $$
  - $$ 6 - (-2) = 8 $$

- Second row:
  - $$ 18 - 14 = 4 $$
  - $$ 18 - 6 = 12 $$
  - $$ -3 - (-4) = 1 $$

- Third row:
  - $$ -15 - 12 = -27 $$
  - $$ 6 - 2 = 4 $$
  - $$ 18 - 0 = 18 $$

Thus,
$$
3B - 2A = \begin{bmatrix} -9 & 8 & 8 \ 4 & 12 & 1 \ -27 & 4 & 18 \end{bmatrix}
$$

**7. Find the additive inverse $$ -A $$:**

Multiply every element in $$ A $$ by $$-1$$:

- First row: 
  - $$ -1 \times 3 = -3 $$
  - $$ -1 \times 2 = -2 $$
  - $$ -1 \times (-1) = 1 $$

- Second row:
  - $$ -1 \times 7 = -7 $$
  - $$ -1 \times 3 = -3 $$
  - $$ -1 \times (-2) = 2 $$

- Third row:
  - $$ -1 \times 6 = -6 $$
  - $$ -1 \times 1 = -1 $$
  - $$ -1 \times 0 = 0 $$

Thus,
$$
-A = \begin{bmatrix} -3 & -2 & 1 \ -7 & -3 & 2 \ -6 & -1 & 0 \end{bmatrix}
$$
**Matrix Operations:** 1. **Addition ($$ A + B $$):** $$ A + B = \begin{bmatrix} 2 & 6 & 1 \ 13 & 9 & -3 \ 1 & 3 & 6 \end{bmatrix} $$ 2. **Subtraction ($$ A - B $$):** $$ A - B = \begin{bmatrix} 4 & -2 & -3 \ 1 & -3 & -1 \ 11 & -1 & -6 \end{bmatrix} $$ 3. **Scalar Multiplication ($$ 2A $$):** $$ 2A = \begin{bmatrix} 6 & 4 & -2 \ 14 & 6 & -4 \ 12 & 2 & 0 \end{bmatrix} $$ 4. **Scalar Multiplication ($$ 3B $$):** $$ 3B = \begin{bmatrix} -3 & 12 & 6 \ 18 & 18 & -3 \ -15 & 6 & 18 \end{bmatrix} $$ 5. **Subtraction ($$ 2A - 3B $$):** $$ 2A - 3B = \begin{bmatrix} 9 & -8 & -8 \ -4 & -12 & -1 \ 27 & -4 & -18 \end{bmatrix} $$ 6. **Subtraction ($$ 3B - 2A $$):** $$ 3B - 2A = \begin{bmatrix} -9 & 8 & 8 \ 4 & 12 & 1 \ -27 & 4 & 18 \end{bmatrix} $$ 7. **Additive Inverse ($$ -A $$):** $$ -A = \begin{bmatrix} -3 & -2 & 1 \ -7 & -3 & 2 \ -6 & -1 & 0 \end{bmatrix} $$

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