Video: Finding Unknowns in Matrix Equations

Find the values of π‘₯ and 𝑦 given the following: [1, 3 and βˆ’2, 1] [2, 0 and π‘₯, 𝑦] = [8, βˆ’9 and βˆ’2, βˆ’3].

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Video Transcript

Find the values of π‘₯ and 𝑦 given the following: the matrix one, three, negative two, one multiplied by the matrix two, zero, π‘₯, 𝑦 equals the matrix eight, negative nine, negative two, negative three.

To solve this for π‘₯ and 𝑦, we can consider how we obtain some of the components and the resultant matrix. Let’s start with the top left component, which is eight. We know that the eight must’ve been obtained by the multiplication of the top row of the first matrix with the left-hand column of the second matrix. That is one times two add three times π‘₯ equals eight or two add three π‘₯ equals eight. We then obtain that three π‘₯ equals six by subtracting two from both sides. And we find that π‘₯ must be equal to two.

Now if we think about how the negative nine was obtained, as this is the top right element, this is the multiplication of the top row of the first matrix with the right-hand column of the second matrix. That is, one times zero add three times 𝑦 must equal negative nine. This simplifies to three 𝑦 equals negative nine. Therefore, 𝑦 must be equal to negative three. We’re then able to verify our answer by checking that these values of π‘₯ and 𝑦 work for the bottom two values in the resultant matrix.

To get the bottom left value of the resultant matrix negative two, we must do the bottom row of the first matrix multiplied by the left-hand column of the second matrix. That is negative two times two add one times π‘₯, which is two, equals negative two. This gives us negative four add two equals negative two, which is true. So our value of π‘₯ is correct. And we can check the bottom right-hand component of the resultant matrix by multiplying the bottom row of the first matrix with the right-hand column of the second matrix. That is negative two times zero add one times negative three equals negative three. That is zero add negative three equals negative three, which is true. So we know our value for 𝑦 is also correct. It’s always good when possible to confirm the values that you found for π‘₯ and 𝑦.

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