Matrix multiplication is not impossible, but it does have specific rules that must be followed. It's important to understand why matrix multiplication might seem "impossible" in certain situations. Here are the key points:

1. Conformability Condition: For two matrices $A$ (of size $m \times n$) and $B$ (of size $p \times q$) to be multiplied, the number of columns in $A$ (which is $n$) must equal the number of rows in $B$ (which is $p$). If this condition is not met, matrix multiplication cannot be performed.

   • For example, if $A$ is a $2 \times 3$ matrix and $B$ is a $4 \times 2$ matrix, you cannot multiply them because the number of columns in $A$ (which is 3) does not equal the number of rows in $B$ (which is 4).

2. Resulting Matrix Dimensions: If matrix multiplication is possible, the resulting matrix $C$ will have dimensions $m \times q$, where $m$ is the number of rows in $A$ and $q$ is the number of columns in $B$.

3. Element-wise Calculation: In matrix multiplication, each element of the resulting matrix is calculated by taking the dot product of the corresponding row from the first matrix and the column from the second matrix. This operation requires that the length of the row vector from matrix $A$ matches the length of the column vector from matrix $B$.

4. Non-Commutative Property: Matrix multiplication is not commutative, meaning that $AB \neq BA$ in most cases. Even if both products are possible, they may yield different results or have different dimensions.

5. Associative Property: Matrix multiplication is associative, meaning $(AB)C = A(BC)$ if the dimensions allow these operations. This property is crucial in many applications.

If you encounter a situation where matrix multiplication seems impossible, it’s likely because the matrices do not meet the conformability condition.

Would you like more details or have any questions about this? Here are some related questions:

1. What are the rules for matrix addition and subtraction?
2. How does the dot product relate to matrix multiplication?
3. Can you explain why matrix multiplication is not commutative?
4. What is the significance of the identity matrix in multiplication?
5. How does matrix multiplication apply in solving systems of equations?
6. What is the role of transpose in matrix multiplication?
7. How do you multiply matrices by hand?
8. Can matrices of any size always be multiplied together?

Tip: When working with matrices, always double-check the dimensions to ensure operations like multiplication are possible!