Given that the two-by-two matrix
with elements five, eight, one, negative eight multiplied by the column matrix with
elements 𝑥 and 𝑦 is equal to the column matrix with elements negative 43 and one,
determine the values of 𝑥 and 𝑦.
We’re asked to find the values of
𝑥 and 𝑦 given a matrix equation. Our matrix equation consists of a
coefficient matrix with elements five, eight, one, and negative eight multiplying a
column matrix of our variables 𝑥 and 𝑦. And this is equal to a column
matrix with elements negative 43 and one. So, in fact, we have a matrix
equation of the form 𝐴𝑥 is equal to 𝑏. And in order to solve for 𝑥 and
𝑦, we can use the fact that for any square nonsingular matrix 𝐴 with inverse 𝐴
inverse, 𝐴 inverse times 𝐴 is equal to 𝐴 times 𝐴 inverse, which is equal to the
identity matrix. And that’s the matrix where the
elements in the leading diagonal all equal one and the rest are equal to zero. And so, if there is a two-by-two
matrix with elements 𝑎, 𝑏, 𝑐, 𝑑, the identity is the matrix with elements one,
zero, zero, one.
Now we can use this in our matrix
equation. Multiplying both sides on the left
with 𝐴 inverse, on our left-hand side we’d have 𝐴 inverse times 𝐴 times 𝑥, which
is actually the identity times 𝑥. And this simplifies to 𝑥 so that
we have 𝑥 equal to 𝐴 inverse times our column matrix 𝑏. And this gives us our solution for
𝑥 and 𝑦. We can apply this to our problem by
first finding the inverse of our two-by-two matrix of coefficients. Recall that an inverse of a
two-by-two matrix with elements 𝑎, 𝑏, 𝑐, 𝑑 is equal to one over 𝑎𝑑 minus 𝑏𝑐
times the matrix with elements 𝑑, negative 𝑏, negative 𝑐, and 𝑎.
Let’s just make some room here and
note that our matrix of coefficients has elements five, eight, one, negative
eight. And if we label these 𝑎, 𝑏, 𝑐,
𝑑, its inverse is given by one over the determinant, which is five times negative
eight minus eight times one times the matrix with elements negative eight, negative
eight, negative one, and five. That is negative one over 48 times
the matrix with elements negative eight, negative eight, negative one, and five. Now if we multiply the left-hand
side of each side of our equation with this inverse, we have our inverse matrix
times the coefficient matrix times the column matrix of variables is equal to the
inverse of the coefficient matrix times the column of constants.
We know that the inverse of the
coefficient matrix times the coefficient matrix itself is equal to the identity. So on our left-hand side, this
simplifies to the column matrix with elements 𝑥, 𝑦. And all we need to do now is to
multiply out our right-hand side. This gives us negative one over 48
times the matrix with negative eight times negative 43 plus negative eight times one
and negative one times negative 43 plus five times one. That is negative one over 48 times
the column matrix with elements 336 and 48. This evaluates to elements negative
seven and negative one. And by equality of matrices, this
gives us 𝑥 is negative seven, 𝑦 is negative one.
And so given the matrix equation
where the coefficient matrix has elements five, eight, one, and negative eight and
the constant column matrix has elements negative 43 and one, the value of 𝑥 is
negative seven and the value of 𝑦 is negative one.