What are the dimensions of the matrix product?

To determine the dimensions of a matrix product, we need to consider the dimensions of the matrices involved in the multiplication. 1. **Matrix Dimensions**: - If matrix \( A \) has dimensions \( m \times n \) (m rows and n columns) and matrix \( B \) has dimensions \( n \times p \) (n rows and p columns), then the product \( AB \) is defined. 2. **Resulting Dimensions**: - The resulting matrix \( AB \) will have dimensions \( m \times p \). This means the number of rows in the resulting matrix is equal to the number of rows in the first matrix \( A \), and the number of columns in the resulting matrix is equal to the number of columns in the second matrix \( B \). 3. **Condition for Multiplication**: - The multiplication is only defined if the number of columns in the first matrix \( A \) (which is \( n \)) is equal to the number of rows in the second matrix \( B \) (also \( n \)). In summary, if \( A \) is \( m \times n \) and \( B \) is \( n \times p \), then the product \( AB \) will have dimensions \( m \times p \).