Video: Using Scalar Multiplication and Addition to Find Unknown Elements in Equal Matrices

Find numbers π‘Ž, 𝑏, and 𝑐 so that π‘Ž [1, 1 and 0, βˆ’1] + 𝑏 [1, 0 and 0, 1] + 𝑐 [0, βˆ’1 and 1, 0] = [1, 0 and βˆ’1, 3].

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Video Transcript

Find numbers π‘Ž, 𝑏, and 𝑐 so that π‘Ž multiplied by the matrix one, one, zero, negative one. Plus 𝑏 multiplied by the matrix one, zero, zero, one. Plus 𝑐 multiplied by the matrix zero, negative one, one, zero. Is equal to the matrix one, zero, negative one, three.

So the first thing I’m gonna do is multiply matrices by either π‘Ž, 𝑏, or 𝑐 on the left-hand side of the equation. Now for scalar multiplication, what we do is we multiply each of our components of our matrix by the scalar value we have. So that’s π‘Ž. So we’re gonna have the matrix π‘Ž, π‘Ž, zero, negative π‘Ž for our first matrix. Then for our second matrix, we’re gonna have 𝑏, zero, zero, 𝑏. That’s cause we have 𝑏 multiplied by one, 𝑏 multiplied by zero, 𝑏 multiplied by zero, and 𝑏 multiplied by one. And then for our final matrix, we’re gonna have zero, negative 𝑐, 𝑐, zero.

So now to find out what the numbers π‘Ž, 𝑏, and 𝑐 are, what we’re gonna do is we’re going to set up some equations. And we can do that by equating the components of our matrices. Cause we know, for instance, that π‘Ž plus 𝑏 plus zero is gonna be equal to one. And that’s because they are corresponding components in our matrices.

So our first equation, like we said, is π‘Ž plus 𝑏 equals one. So now if we take a look at our next equation, we’ve got π‘Ž plus zero plus negative 𝑐, well gonna be equal to zero. So therefore, we’d have π‘Ž minus 𝑐 is equal to zero. So then we’re gonna have zero plus zero plus 𝑐 is equal to negative one. So therefore, we’ve got that 𝑐 is equal to negative one. And then, finally, what we’ve got is negative π‘Ž plus 𝑏 is equal to three.

Okay, great. So that’s our equations. So let’s use them to find out π‘Ž, 𝑏, and 𝑐. Well, we already know that 𝑐 is equal to negative one. So now what we’re going to find next? Well, if we have π‘Ž minus 𝑐 is equal to zero, and then we add 𝑐 to each side of our equation, we’re gonna get π‘Ž is equal to 𝑐. So therefore, if we know that π‘Ž is equal to 𝑐, then π‘Ž is gonna also be negative one.

So now all we need to do is find out 𝑏. Well, if we utilised the first equation, we could’ve done the first or the fourth. But we’re gonna go for the first. Then what we’re gonna get is negative one β€” and that’s cause π‘Ž is equal to negative one β€” plus 𝑏 is equal to one. So then if we add one to each side of the equation, we’re gonna get 𝑏 is equal to two.

So therefore, we solved the problem cause we found the three numbers π‘Ž, 𝑏, and 𝑐. And they are π‘Ž is equal to negative one, 𝑏 is equal to two, and 𝑐 is equal to negative one.

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