### 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 Iβm 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 thatβs π. So weβre gonna have the matrix π,
π, zero, negative π for our first matrix. Then for our second matrix, weβre
gonna have π, zero, zero, π. Thatβs 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,
weβre gonna have zero, negative π, π, zero.

So now to find out what the numbers
π, π, and π are, what weβre gonna do is weβre 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 thatβs 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, weβve got π plus zero plus negative π, well gonna be equal to
zero. So therefore, weβd have π minus π
is equal to zero. So then weβre gonna have zero plus
zero plus π is equal to negative one. So therefore, weβve got that π is
equal to negative one. And then, finally, what weβve got
is negative π plus π is equal to three.

Okay, great. So thatβs our equations. So letβs use them to find out π,
π, and π. Well, we already know that π is
equal to negative one. So now what weβre 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, weβre 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 couldβve done the first or the fourth. But weβre gonna go for the
first. Then what weβre gonna get is
negative one β and thatβs 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, weβre 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.