# Question Video: Using Scalar Multiplication and Addition to Find Unknown Elements in Equal Matrices Mathematics • 10th Grade

Find numbers π, π, and π so that π [1, 1 and 0, β1] + π [1, 0 and 0, 1] + π [0, β1 and 1, 0] = [1, 0 and β1, 3].

02:56

### Video Transcript

Find numbers π, π, and π so that π multiplied by the matrix one, one, zero, negative one. Plus π multiplied by the matrix one, zero, zero, one. Plus π multiplied by the matrix zero, negative one, one, zero. Is equal to the matrix one, zero, negative one, three.

So the first thing Iβm gonna do is multiply matrices by either π, π, or π 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 thatβs π. So weβre gonna have the matrix π, π, zero, negative π for our first matrix. Then for our second matrix, weβre gonna have π, zero, zero, π. Thatβs cause we have π multiplied by one, π multiplied by zero, π multiplied by zero, and π multiplied by one. And then for our final matrix, weβre gonna have zero, negative π, π, zero.

So now to find out what the numbers π, π, and π are, what weβre gonna do is weβ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 plus π plus zero is gonna be equal to one. And thatβs because they are corresponding components in our matrices.

So our first equation, like we said, is π plus π equals one. So now if we take a look at our next equation, weβve got π plus zero plus negative π, well gonna be equal to zero. So therefore, weβd have π minus π is equal to zero. So then weβre gonna have zero plus zero plus π is equal to negative one. So therefore, weβve got that π is equal to negative one. And then, finally, what weβve got is negative π plus π is equal to three.

Okay, great. So thatβs our equations. So letβs use them to find out π, π, and π. Well, we already know that π is equal to negative one. So now what weβre going to find next? Well, if we have π minus π is equal to zero, and then we add π to each side of our equation, weβre gonna get π is equal to π. So therefore, if we know that π is equal to π, then π is gonna also be negative one.

So now all we need to do is find out π. Well, if we utilised the first equation, we couldβve done the first or the fourth. But weβre gonna go for the first. Then what weβre gonna get is negative one β and thatβs cause π is equal to negative one β plus π is equal to one. So then if we add one to each side of the equation, weβre gonna get π is equal to two.

So therefore, we solved the problem cause we found the three numbers π, π, and π. And they are π is equal to negative one, π is equal to two, and π is equal to negative one.