Question Video: Operations on Matrices Mathematics

Video Transcript

Consider the matrices 𝑍 equals the matrix negative seven, four, negative one, one, and 𝐴 equals a matrix seven, four, three, three. Is it true that negative seven multiplied by the matrix 𝑍 plus the matrix 𝐴 is equal to negative seven multiplied by the matrix 𝐴 minus seven multiplied by the matrix 𝑍?

Well, the first thing we need to work out in this question is what is matrix 𝑍 plus matrix A? So, to work this out, what we need to do is we need to add the corresponding components of each of our matrices. So, for example, if you want the top-left component, what we’re gonna do is we’re gonna add negative seven and seven. And then, if we do this for the rest of the components, we’re gonna have four add four, negative one add three, and one add three.

So, this is gonna leave us with the matrix zero, eight, two, four. So, now what we want to do is we want to multiply this by negative seven. And to do that, what we do is we multiply each component by negative seven, so we get zero multiplied by negative seven, eight multiplied by negative seven, two multiplied by negative seven, and four multiplied by negative seven. And this leaves us with the matrix zero, negative 56, negative 14, negative 28.

So, great, that’s the left-hand side of our equation worked out. Now, we need to work out our right-hand side of the equation. And to do this, we need to find out what negative seven multiplied by the matrix 𝐴 minus seven multiplied by the matrix 𝑍 is. So, when we multiply matrix 𝐴 by negative seven, we’re getting negative 49, and we got that because negative seven multiplied by seven is negative 49, negative 28, negative 21, and negative 21. So, that’s our negative seven multiplied by the matrix 𝐴.

Now, what we want to do is work out what seven multiplied by the matrix 𝑍 is. And when I do this, what I get is negative 49, 28, negative seven, seven is our matrix. Again, we did this by multiplying each of our components by seven. So, now we move on because what we want to work out is what negative seven multiplied by the matrix 𝐴 minus seven multiplied by the matrix 𝑍 is. And to do that, what we do is we subtract the components that we found for seven multiplied by matrix 𝑍 by all the ones we found for negative seven multiplied by matrix 𝐴. So, we get negative 49 minus negative 49, negative 28 minus 28, negative 21 minus negative seven, and negative 21 minus seven.

So then, when we carry out these calculations, what we get is the matrix zero, negative 56, negative 14, negative 28. So, we found out that the left-hand side is equal to the right-hand side. So, in answer to our question, we can say that it is true that negative seven multiplied by the matrix 𝑍 plus 𝐴 is equal to negative seven multiplied by the matrix 𝐴 minus seven multiplied by the matrix 𝑍. So, we’ve done that, and we proved it to be correct.

But we could’ve worked out using another quicker method. And that method would’ve been to use the distributive properties of multiplication. And that’s because if we had 𝑎 multiplied by 𝑏 plus 𝑐, this is equal to 𝑎 multiplied by 𝑏 plus 𝑎 multiplied by 𝑐. So, in our example, the negative seven multiplied by the matrix 𝑍 would be the bit which is negative seven multiplied by matrix 𝑍. And the negative seven multiplied by matrix 𝐴 would be the first part of our right-hand side, which is negative seven multiplied by matrix 𝐴. So, that would’ve been another way that we could’ve proved that the answer was yes.