Let's analyze the problem to determine if the matrix operation $A + B + C$ is possible.

Step 1: Determine the Dimensions

First, let’s find the dimensions of each matrix.

• Matrix A is given as: $A = \begin{bmatrix} -4 & -4 \\ 1 & -3 \\ -8 & 9 \end{bmatrix}$ Dimensions: $3 \times 2$ (3 rows and 2 columns)

• Matrix B is given as: $B = \begin{bmatrix} 1 & -3 \\ -5 & -4 \\ 4 & 3 \end{bmatrix}$ Dimensions: $3 \times 2$ (3 rows and 2 columns)

• Matrix C is given as: $C = \begin{bmatrix} -6 & -1 \\ -10 & -9 \\ -9 & -8 \end{bmatrix}$ Dimensions: $3 \times 2$ (3 rows and 2 columns)

Step 2: Check if Addition is Possible

For matrix addition to be possible, all matrices involved must have the same dimensions.

• Matrices A, B, and C all have dimensions of $3 \times 2$.

Since they all have the same dimensions, addition is possible.

Step 3: Perform the Addition

Now, we will add the corresponding elements from each matrix.

$A + B + C = \begin{bmatrix} -4 & -4 \\ 1 & -3 \\ -8 & 9 \end{bmatrix} + \begin{bmatrix} 1 & -3 \\ -5 & -4 \\ 4 & 3 \end{bmatrix} + \begin{bmatrix} -6 & -1 \\ -10 & -9 \\ -9 & -8 \end{bmatrix}$

Calculating each element:

1. First row:

   • $(-4 + 1 - 6) = -9$
   • $(-4 + (-3) - 1) = -8$

2. Second row:

   • $(1 + (-5) - 10) = -14$
   • $(-3 + (-4) - 9) = -16$

3. Third row:

   • $(-8 + 4 - 9) = -13$
   • $(9 + 3 - 8) = 4$

Thus:

$A + B + C = \begin{bmatrix} -9 & -8 \\ -14 & -16 \\ -13 & 4 \end{bmatrix}$

Final Answer

$A + B + C = \begin{bmatrix} -9 & -8 \\ -14 & -16 \\ -13 & 4 \end{bmatrix}$

Would you like more details or have any questions?

Related Questions:

1. How do you determine if two matrices can be multiplied?
2. What are the properties of matrix addition?
3. How can you find the transpose of a matrix?
4. How do you perform scalar multiplication with matrices?
5. What is the identity matrix and how is it used?

Tip:

Always make sure that the matrices involved in addition have the exact same dimensions before attempting to add them.