# Question Video: Distributing Matrix Multiplication over Addition Mathematics

Given three matrices 𝐴, 𝐵, and 𝐶, which of the following is equal to 𝐴(𝐵 + 𝐶)? [A] 𝐴𝐵 + 𝐶 [B] 𝐴𝐵 + 𝐴𝐶 [C] 𝐵𝐴 + 𝐶𝐴 [D] 𝐵𝐴 + 𝐶 [E] 𝐵 + 𝐴𝐶

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### 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.

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