Video Transcript
Given three matrices 𝐴, 𝐵, and
𝐶, which of the following is equal to 𝐴 multiplied by 𝐵 plus 𝐶? Is it (A) 𝐴𝐵 plus 𝐶, (B) 𝐴𝐵
plus 𝐴𝐶, (C) 𝐵𝐴 plus 𝐶𝐴, (D) 𝐵𝐴 plus 𝐶, or (E) 𝐵 plus 𝐴𝐶?
In order to answer this question,
we need to use the distributive property of matrices. We can distribute matrices in a
similar way to how we distribute real numbers. Multiplying matrix 𝐴 by matrix 𝐵
plus 𝐶 is equal to matrix 𝐴𝐵 plus matrix 𝐴𝐶. It is important to note though that
if the parentheses came first, we were multiplying 𝐵 plus 𝐶 by 𝐴, then our answer
would be 𝐵𝐴 plus 𝐶𝐴. If the matrix 𝐴 is distributed
from the left side, we must ensure that the product in the resulting sum has 𝐴 on
the left. In the same way, if matrix 𝐴 is
distributed from the right side, each product in the resulting sum must have 𝐴 on
the right. We can therefore see that the
correct answer is option (B). 𝐴 multiplied by 𝐵 plus 𝐶 is
equal to 𝐴𝐵 plus 𝐴𝐶.
It is important to remember that
when performing matrix addition and matrix multiplication, the order of each matrix
is key. In order to add matrix 𝐵 and 𝐶,
they must have the same order. To perform matrix multiplication,
the number of columns in matrix 𝐴 must be equal to the number of rows in matrix 𝐵
and 𝐶.
