Video: Finding the Matrix That Satisfies a Given Matrix Equation Using Operations on Matrices

Solve for matrix ๐‘‹ in the matrix equation 3๐‘‹ + ๐ต = ๐ถ, where ๐ต and ๐ถ are as follows: ๐ต = the matrix 5, 7, โˆ’10, โˆ’8 and ๐ถ = the matrix 8, โˆ’2, 2, 7.

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Video Transcript

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

Okay, first of all, what weโ€™re gonna have a look at is the equation three ๐‘‹ plus ๐ต equals ๐ถ. Well, actually, what weโ€™re trying to do is weโ€™re 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, weโ€™re gonna subtract ๐ต from each side. So we get three ๐‘‹ is equal to ๐ถ minus ๐ต. Then, if we divide both sides by three, we get that matrix ๐‘‹ is equal to a third multiplied by matrix ๐ถ minus matrix ๐ต.

Okay, now that we know how we can find ๐‘‹, what weโ€™re gonna start with- weโ€™re actually gonna start with matrix ๐ถ minus matrix ๐ต. Okay, to find matrix ๐ถ minus matrix ๐ต, weโ€™re 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 weโ€™re gonna start with the top left element and thatโ€™s gonna be eight minus five. And thatโ€™s 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 theyโ€™re corresponding elements. And then, weโ€™re gonna have two minus negative 10. Okay and then for our final element, weโ€™re actually gonna have seven minus negative eight.

Okay, great! Weโ€™ve got all of the elements filled in. So now, we just calculate the values. And we can actually have the matrix solution of matrix ๐ถ minus matrix ๐ต. So therefore, we can say that the matrix ๐ถ minus matrix ๐ต is equal to the matrix three, negative nine, 12, 15. Okay, great! But is this ๐‘‹? Well, no, weโ€™ve 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 ๐ถ minus matrix ๐ต. 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 itโ€™s a third, all weโ€™re 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.

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