Let's work through this matrix multiplication problem step by step.

Problem Statement

We are given two matrices: $A = \begin{bmatrix} 3 & 0 & 1 \\ 5 & 5 & 6 \end{bmatrix}$ $B = \begin{bmatrix} 4 & 7 \\ 7 & 7 \end{bmatrix}$

We need to calculate $A \times B$.

Step 1: Confirm Matrix Dimensions

Matrix $A$ has dimensions $2 \times 3$ (2 rows, 3 columns), and matrix $B$ has dimensions $2 \times 2$ (2 rows, 2 columns).

For matrix multiplication to be possible, the number of columns in $A$ must equal the number of rows in $B$. Since $A$ has 3 columns and $B$ has 2 rows, the multiplication $A \times B$ is not defined because the inner dimensions do not match.

Conclusion

Matrix multiplication $A \times B$ is undefined in this case because the dimensions do not align correctly for multiplication.

Would you like to ask about a related concept, or do you have any questions?

Here are 5 related questions to deepen understanding:

1. What are the rules for matrix multiplication regarding dimensions?
2. How can we verify if two matrices can be multiplied?
3. What does it mean if matrix multiplication is undefined?
4. How is matrix multiplication different from element-wise multiplication?
5. Can matrix multiplication be commutative?

Tip: Always verify the dimensions of matrices before performing operations like multiplication.