### Video Transcript

Solve for matrix ๐ in the matrix equation three ๐ plus ๐ต equals ๐ถ, where ๐ต and ๐ถ are as follows: ๐ต is equal to the matrix five, seven, negative 10, negative eight and ๐ถ is equal to the matrix eight, negative two, two, seven.

Okay, first of all, what weโre gonna have a look at is the equation three ๐ plus ๐ต equals ๐ถ. Well, actually, what weโre trying to do is weโre trying to solve matrix ๐. So what we want to do is rearrange this equation to have ๐ as the subject. So first of all, weโre gonna subtract ๐ต from each side. So we get three ๐ is equal to ๐ถ minus ๐ต. Then, if we divide both sides by three, we get that matrix ๐ is equal to a third multiplied by matrix ๐ถ minus matrix ๐ต.

Okay, now that we know how we can find ๐, what weโre gonna start with- weโre actually gonna start with matrix ๐ถ minus matrix ๐ต. Okay, to find matrix ๐ถ minus matrix ๐ต, weโre gonna have the matrix eight, negative two, two, seven minus the matrix five, seven, negative 10, negative eight.

And in order to subtract two matrices, what we actually do is we subtract the corresponding elements. So weโre gonna start with the top left element and thatโs gonna be eight minus five. And thatโs because the eight and the five are the corresponding elements of our two matrices. Then, our top right element will be negative two minus seven, again cause theyโre corresponding elements. And then, weโre gonna have two minus negative 10. Okay and then for our final element, weโre actually gonna have seven minus negative eight.

Okay, great! Weโve got all of the elements filled in. So now, we just calculate the values. And we can actually have the matrix solution of matrix ๐ถ minus matrix ๐ต. So therefore, we can say that the matrix ๐ถ minus matrix ๐ต is equal to the matrix three, negative nine, 12, 15. Okay, great! But is this ๐? Well, no, weโve got one more stage left to do.

Okay, for this final stage, we can remind ourselves that matrix ๐ is equal to a third multiplied by matrix ๐ถ minus matrix ๐ต. So therefore, we can say that matrix ๐ is equal to a third multiplied by the matrix three, negative nine, 12, 15.

So therefore, to get our final answer for matrix ๐, what we need to do is we actually need to multiply each element of our matrix by a third. And actually to do that, because itโs a third, all weโre gonna do is divide each of them by three. And therefore, by dividing each element by three, we can say that the matrix ๐ is equal to the matrix one, negative three, four, five.