The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, if the products are defined.
A is 4 × 2, B is 2 × 4.
To find the sizes of the product AB, we can multiply the number of columns in A with the number of rows in B.
A has 2 columns and B has 2 rows.
Therefore, the product AB will have a size of 4 × 4.
To find the sizes of the product BA, we can multiply the number of columns in B with the number of rows in A.
B has 4 columns and A has 4 rows.
Therefore, the product BA will have a size of 2 × 2.
To find the sizes of the product AB and BA, we need to know about matrix multiplication rules.
In matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. Let's denote the sizes as m × n for matrix A (m rows and n columns) and p × q for matrix B (p rows and q columns).
Given that matrix A is 4 × 2 (4 rows and 2 columns) and matrix B is 2 × 4 (2 rows and 4 columns), we can determine the sizes of the product matrices.
To calculate AB, the number of columns in A (n = 2) must be equal to the number of rows in B (p = 2). Since n = p = 2, the product AB is defined. The resulting matrix will have the dimensions of m × q, which are 4 × 4.
Therefore, the size of the product AB is 4 × 4.
On the other hand, to calculate BA, the number of columns in B (q = 4) must be equal to the number of rows in A (m = 4). Since q = m = 4, the product BA is defined. The resulting matrix will have the dimensions of p × n, which are 2 × 2.
Therefore, the size of the product BA is 2 × 2.