determine the dimensions of each matrix product
A X B
2X3 3X5
G X C
4X7 7X1
If you notice that for AB, it's 2x3 3x5 where the number of columns of A must match the number of rows of B. After multiplication, they cancel out and the product AB becomes a 2x5.
Same goes for GC.
To determine the dimensions of a matrix product, you need to consider the number of rows and columns in each matrix involved in the multiplication.
Let's calculate the dimensions for each matrix product:
1) A × B:
Matrix A has dimensions 2 × 3, meaning it has 2 rows and 3 columns.
Matrix B has dimensions 3 × 5, meaning it has 3 rows and 5 columns.
To multiply these matrices, the number of columns in the first matrix (A) must be equal to the number of rows in the second matrix (B). In this case, since A has 3 columns and B has 3 rows, they match.
The resulting matrix will have dimensions where the number of rows matches the number of rows in the first matrix and the number of columns matches the number of columns in the second matrix. Therefore, the product A × B will have dimensions 2 × 5 (2 rows and 5 columns).
2) G × C:
Matrix G has dimensions 4 × 7, meaning it has 4 rows and 7 columns.
Matrix C has dimensions 7 × 1, meaning it has 7 rows and 1 column.
Using the same logic, since G has 7 columns and C has 7 rows, they match.
The resulting matrix will have dimensions where the number of rows matches the number of rows in the first matrix and the number of columns matches the number of columns in the second matrix. Therefore, the product G × C will have dimensions 4 × 1 (4 rows and 1 column).