Given that matrix 𝐴 is equal to zero, three, negative two, one, six, negative one, matrix 𝐵 is equal to negative five, negative six, one, four, and matrix 𝐶 is equal to negative three, zero, four, negative two, is it true that 𝐴𝐵 multiplied by 𝐶 is equal to 𝐴 multiplied by 𝐵𝐶?
Let’s begin by considering 𝐴𝐵 multiplied by 𝐶, where we firstly need to calculate the matrix 𝐴𝐵. When multiplying two matrices, we need to ensure that the number of columns in the first matrix is equal to the number of rows in the second matrix. As long as this is the case, we then multiply the elements of each row in the first matrix by the elements of each column in the second matrix. We then add the products. The top-left element in matrix 𝐴𝐵 is therefore equal to zero multiplied by negative five plus three multiplied by one. For the top-right element, we multiply the first row in matrix 𝐴 by the second column in matrix 𝐵. We have zero multiplied by negative six plus three multiplied by four.
We then repeat this for the second row of matrix 𝐴. Finally, we multiply the third row in matrix 𝐴 by each of the columns in matrix 𝐵. We can then work out each of these individual calculations. Matrix 𝐴𝐵 is therefore equal to three, 12, 11, 16, negative 31, negative 40. Our next step is to multiply this matrix by matrix 𝐶 negative three, zero, four, negative two. This is equal to the three-by-two matrix 39, negative 24, 31, negative 32, negative 67, 80.
We now need to consider the right-hand side of the equation, 𝐴 multiplied by 𝐵𝐶. We begin by calculating 𝐵 multiplied by 𝐶. We need to multiply the two-by-two matrices negative five, negative six, one, four and negative three, zero, four, negative two. This is equal to the two-by-two matrix negative nine, 12, 13, negative eight. We then need to multiply this matrix by the three-by-two matrix zero, three, negative two, one, six, negative one.
As matrix multiplication is not commutative, the order matters. Matrix 𝐴 must come first. Multiplying these two matrices gives us 39, negative 24, 31, negative 32, negative 67, 80. Our two matrices 𝐴𝐵 multiplied by 𝐶 and 𝐴 multiplied by 𝐵𝐶 are identical. We can therefore conclude that when multiplying the three matrices, the statement is true.