Question Video: Distributing Matrix Multiplication over Addition Mathematics

Given three matrices 𝐴, 𝐡, and 𝐢, which of the following is equal to 𝐴(𝐡 + 𝐢)? [A] 𝐴𝐡 + 𝐢 [B] 𝐴𝐡 + 𝐴𝐢 [C] 𝐡𝐴 + 𝐢𝐴 [D] 𝐡𝐴 + 𝐢 [E] 𝐡 + 𝐴𝐢

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Video Transcript

Given three matrices 𝐴, 𝐡, and 𝐢, which of the following is equal to 𝐴 multiplied by 𝐡 plus 𝐢? Is it (A) 𝐴𝐡 plus 𝐢, (B) 𝐴𝐡 plus 𝐴𝐢, (C) 𝐡𝐴 plus 𝐢𝐴, (D) 𝐡𝐴 plus 𝐢, or (E) 𝐡 plus 𝐴𝐢?

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 𝐡 plus 𝐢 is equal to matrix 𝐴𝐡 plus matrix 𝐴𝐢. It is important to note though that if the parentheses came first, we were multiplying 𝐡 plus 𝐢 by 𝐴, then our answer would be 𝐡𝐴 plus 𝐢𝐴. 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 𝐡 plus 𝐢 is equal to 𝐴𝐡 plus 𝐴𝐢.

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 𝐡 and 𝐢, 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 𝐡 and 𝐢.

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