Video Transcript
Write down the set of simultaneous
equations that could be solved using the given matrix equation, that is, the
three-by-three matrix with elements one, negative two, negative four, one, zero,
one, three, four, negative eight multiplied by the column matrix 𝑝, 𝑞, 𝑟 is equal
to the column matrix 11, six, 10.
We’re given a matrix equation with
a three-by-three coefficient matrix and a three-by-one column matrix of variables on
the left-hand side and a three-by-one column matrix of constants on the right-hand
side. The fact that our coefficient
matrix has three rows means that we’ll have three equations in our system of
equations. And since the coefficient matrix
also has three columns, this confirms that our system has three variables, which we
can see in our column matrix of variables with variables 𝑝, 𝑞, and 𝑟.
In fact, the elements in each row
of our coefficient matrix are the coefficients in one of our equations of the
variables 𝑝, 𝑞, and 𝑟. And all we need to do to form the
first equation is to carry out the first row times column operation in matrix
multiplication, that is, one times 𝑝 plus negative 𝑞 times two plus negative four
times 𝑟. And this will equal our constant 11
in the first row of the matrix on the right-hand side. And so our first equation is 𝑝
minus two 𝑞 minus four 𝑟 is equal to 11.
So for our second equation, let’s
do the same thing with the second row of the coefficient matrix. In this case, we have one times 𝑝
plus zero times 𝑞 plus one times 𝑟 is equal to six. That is, 𝑝 plus 𝑟 is equal to
six. And finally, with the third row of
our coefficient matrix, we have three times 𝑝 plus four times 𝑞 plus negative
eight times 𝑟 is equal to 10 on the right-hand side. That is, three 𝑝 plus four 𝑞
minus eight 𝑟 is equal to 10.
And with these three equations,
then, we can write down the set of simultaneous equations that could be solved using
the given matrix. And these are 𝑝 minus two 𝑞 minus
four 𝑟 is equal to 11, 𝑝 plus 𝑟 is equal to six, and three 𝑝 plus four 𝑞 minus
eight 𝑟 is equal to 10. We can check that we do have the
correct system of equations by comparing the coefficients of the variables in each
equation with the relevant row in the original matrix. So for example, in the first
equation, the coefficients of 𝑝, 𝑞, and 𝑟 are one, negative two, and negative
four, respectively, which correspond to the elements in the first row of our
coefficient matrix and with the constant 11 on the right-hand side. And we can do the same thing for
the second two equations.
