Video: Distributing Matrix Multiplication over Addition

Suppose that 𝐴 = [1, −3 and −4, 2], 𝐵 = [2, 0 and 1, −1], and 𝐶 = [0, 1 and −3, 0]. Find 𝐴𝐵. Find 𝐴𝐶. Find 𝐴(2𝐵 + 7𝐶). Express 𝐴(2𝐵 + 7𝐶) in terms of 𝐴𝐵 and 𝐴𝐶.

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Video Transcript

Suppose matrix 𝐴 is equal to one, negative three, negative four, two; matrix 𝐵 is equal to two, zero, one, negative one; and matrix 𝐶 is equal to zero, one, negative three, zero. There are four parts to this question. Find matrix 𝐴𝐵. Find matrix 𝐴𝐶. Find 𝐴 multiplied by two 𝐵 plus seven 𝐶. And express 𝐴 multiplied by two 𝐵 plus seven 𝐶 in terms of 𝐴𝐵 and 𝐴𝐶.

In order to multiply matrix 𝐴 by matrix 𝐵, we need to multiply all of the elements in the rows of matrix 𝐴 by the columns of matrix 𝐵. One multiplied by two plus negative three multiplied by one is equal to negative one. Repeating this for the other rows and columns gives us the elements three, negative six, and negative two. Matrix 𝐴𝐵 is equal to negative one, three, negative six, negative two.

To work out matrix 𝐴𝐶, we multiply one, negative three, negative four, two by zero, one, negative three, zero. This gives us the elements nine, one, negative six, and negative four. This is the matrix 𝐴𝐶.

In the third part of our question, we begin by multiplying matrix 𝐵 by the scalar or constant two and matrix 𝐶 by the scalar seven. When multiplying a matrix by a scalar, we simply multiply each of the elements by that scalar. This means that two 𝐵 is equal to four, zero, two, negative two. In the same way, seven 𝐶 is equal to zero, seven, negative 21, zero.

Next, we need to add these two matrices. We do this by adding the elements in corresponding positions in each matrix. Four plus zero is equal to four. Repeating this for the other elements gives us the matrix four, seven, negative 19, negative two.

Finally, we need to multiply this matrix by matrix 𝐴. The order here is important. We must multiply matrix 𝐴 by the matrix four, seven, negative 19, negative two. This gives us the elements 61, 13, negative 54, and negative 32. 𝐴 multiplied by two 𝐵 plus seven 𝐶 is equal to 61, 13, negative 54, negative 32.

In the final part of this question, we can use the distributive property of matrix multiplication. We can multiply matrix 𝐴 by two 𝐵 and then add matrix 𝐴 multiplied by seven 𝐶. This gives us one, negative three, negative four, two multiplied by four, zero, two, negative two plus one, negative three, negative four, two multiplied by zero, seven, negative 21, zero. The first product gives us negative two, six, negative 12, negative four. The second product gives us 63, seven, negative 42, negative 28.

We might be tempted to simply add these matrices. However, we were asked to give our answer in terms of 𝐴𝐵 and 𝐴𝐶. We notice that our first matrix negative two, six, negative 12, negative four is two times matrix 𝐴𝐵. We also notice that the second matrix 63, seven, negative 42, negative 28 is seven times matrix 𝐴𝐶. This means that matrix 𝐴 multiplied by two 𝐵 plus seven 𝐶 is equal to two multiplied by matrix 𝐴𝐵 plus seven multiplied by matrix 𝐴𝐶.

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