Question Video: Solving a Matrix Equation by Finding the Inverse

Given that π‘₯ Γ— [βˆ’2, 0 and βˆ’3, βˆ’5] = [14, 0 and 21, 35], find the value of π‘₯.

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Video Transcript

Given that π‘₯ times the two-by-two matrix negative two, zero, negative three, negative five is equal to the two-by-two matrix 14, zero, 21, 35, find the value of π‘₯.

We’re given an equation involving two matrices and π‘₯, and we need to determine the value of π‘₯. First, we need to notice by looking at our equation, π‘₯ is a number. It’s not a matrix. This is because π‘₯ is given in lowercase. Another reason we can see this is we’re asked to find the value of π‘₯. We’re only asked to do this when π‘₯ represents a number. So in our equation, when we multiply our matrix by 𝐴, this is scalar multiplication of a matrix. So the first thing we’ll do is start by π‘₯ multiplied by our matrix and apply scalar multiplication to rewrite this in a different form.

To do this, we need to recall that when we multiply a matrix by a scalar, we multiply every single entry inside of our matrix by our scalar. In this case, we need to multiply all of our entries by π‘₯. By doing this, we get the two-by-two matrix of entries π‘₯ times negative two, π‘₯ times zero, π‘₯ times negative three, and π‘₯ times negative five. And of course, we can evaluate or simplify all of these entries. Doing this, we get the two-by-two matrix negative two π‘₯, zero, negative three π‘₯, negative five π‘₯.

But remember, in the question, we’re told that this matrix is exactly equal to the two-by-two matrix 14, zero, 21, 35. So we’ve now shown that these two matrices are equal. To find our value of π‘₯, we’re going to need to recall what it means for two matrices to be equal. Recall that we say that two matrices are equal if every single entry in both of our matrices are equal and they have the same order. Of course, both of these are two-by-two matrices. So all we need to do is check that all of their entries are equal.

Doing this, we get a series of equations. By equating the entries in row one and column one, we get 14 should be equal to negative two π‘₯. By equating the entries in row one and column two of our matrices, we get that zero should be equal to zero. By equating the entries in row two and column one of both of our matrices, we get that 21 should be equal to negative three π‘₯. And finally, by equating the entries in row two and column two of our matrices, we get that 35 should be equal to negative five π‘₯. This gives us a system of equations we need to solve. In fact, all of these are linear equations, so we could call this a system of linear equations.

Firstly, we can see that the second equation is true for all values of π‘₯. Next, we can just solve all three of the remaining linear equations. We’ll divide the first one through by negative two, the second one through by negative three, and the third one through by negative five. And by doing this, we can see that all of these are solved by π‘₯ being equal to negative seven. This means we were able to show that the value of π‘₯ should be negative seven. However, it’s also worth pointing out we can confirm this answer either by substituting the value of π‘₯ is equal to negative seven into our matrix or substituting π‘₯ is equal to negative seven into our original equation. We could then verify that this gives us the correct solution to the equation.

Therefore, given that π‘₯ times the two-by-two matrix negative two, zero, negative three, negative five was equal to the two-by-two matrix 14, zero, 21, 35, we were able to show that the value of π‘₯ must be equal to negative seven.

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