Let's go through each of the matrix operations to determine if they are possible and, if so, calculate the resulting matrix size.

(a) $AB$

• $A$ is a $6 \times 2$ matrix.
• $B$ is a $2 \times 3$ matrix.

To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Here, the number of columns of $A$ (which is 2) equals the number of rows of $B$ (which is 2), so the multiplication is possible.

The size of the resulting matrix is determined by the number of rows of the first matrix and the number of columns of the second matrix. So the size of $AB$ will be: $\text{size of } AB = 6 \times 3$

(b) $BC$

• $B$ is a $2 \times 3$ matrix.
• $C$ is a $3 \times 4$ matrix.

Again, the number of columns in $B$ (3) matches the number of rows in $C$ (3), so the multiplication is possible.

The resulting matrix will have the number of rows of $B$ and the number of columns of $C$. Therefore, the size of $BC$ is: $\text{size of } BC = 2 \times 4$

(c) $AC$

• $A$ is a $6 \times 2$ matrix.
• $C$ is a $3 \times 4$ matrix.

Here, the number of columns of $A$ (2) does not match the number of rows of $C$ (3), so this multiplication is not possible.

Thus, the answer for this operation is: $\text{size of } AC = \text{DNE} \ (\text{Does Not Exist})$

(d) $(AB)C$

We already determined that $AB$ is possible and results in a $6 \times 3$ matrix. Now, we need to check if $(AB)C$ is possible.

• $AB$ is a $6 \times 3$ matrix.
• $C$ is a $3 \times 4$ matrix.

Since the number of columns of $AB$ (3) matches the number of rows of $C$ (3), this multiplication is possible.

The resulting matrix will have the number of rows of $AB$ and the number of columns of $C$. Therefore, the size of $(AB)C$ is: $\text{size of } (AB)C = 6 \times 4$

Final Answers:

(a) $AB$ → size = $6 \times 3$

(b) $BC$ → size = $2 \times 4$

(c) $AC$ → size = DNE

(d) $(AB)C$ → size = $6 \times 4$

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Would you like more details or have any other questions?

Here are five related questions to expand your understanding:

1. What are the rules for matrix multiplication involving square matrices?
2. How would you compute the actual values of $AB$ and $BC$ if given matrix entries?
3. What happens if you try to multiply matrices of incompatible sizes?
4. What is the significance of matrix dimensions in linear transformations?
5. How does matrix multiplication relate to systems of equations?

Tip: Always check the compatibility of dimensions before multiplying matrices to avoid unnecessary calculations!