Solve the system of linear
equations negative 𝑥 plus 𝑦 plus 𝑧 equals eight, negative two 𝑥 plus 𝑦 minus 𝑧
equals negative five, and six 𝑥 minus three 𝑦 equals negative six using the
inverse of a matrix.
First, we need to represent our
system of equations as a matrix equation. Our matrix equation will be of the
form 𝐴𝑋 equals 𝐵, where 𝐴 is the coefficient matrix, 𝑋 is the variable matrix,
and 𝐵 is the constant matrix.
Let’s suppose we had the system of
linear equations 𝑎 one 𝑥 plus 𝑏 one 𝑦 plus 𝑐 one 𝑧 equals 𝑑 one, 𝑎 two 𝑥
plus 𝑏 two 𝑦 plus 𝑐 two 𝑧 equals 𝑑 two, and 𝑎 three 𝑥 plus 𝑏 three 𝑦 plus
𝑐 three 𝑧 equals 𝑑 three. We can represent this in matrix
form by putting the coefficients of the 𝑥-, 𝑦-, and 𝑧-terms in the coefficient
matrix. The variable matrix consists of our
three variables 𝑥, 𝑦, and 𝑧. And the constant matrix consists of
the constants which our three equations are equal to.
We may notice that our third
equation doesn’t have a 𝑧-term. This just means that the
coefficient of the 𝑧-term is zero, So we can write this in if we want to. Our equation in matrix form is
therefore negative one, one, one, negative two, one, negative one, six, negative
three, zero multiplied by 𝑥, 𝑦, 𝑧, is equal to eight, negative five, negative
Now we have our system in the form
𝐴𝑋 equals 𝐵. We need to use the inverse of a
matrix to solve it. Since we’re trying to find 𝑋, we
can multiply both sides of the equation by the inverse of 𝐴 on the left. Since 𝐴 inverse multiplied by 𝐴
is just the identity matrix, the left-hand side just becomes 𝑋. We now have that 𝑋 is equal to 𝐴
inverse multiplied by 𝐵.
Our next step is to find the
inverse of the matrix 𝐴. Let’s clear some space. We need to find the inverse of our
matrix 𝐴. And we can do this using the
adjoint method, which is used to find the inverse of three-by-three matrices. There are five steps to finding the
inverse of a matrix using the adjoint method. The first of these steps is finding
the determinant of the matrix. This will also tell us whether the
matrix is invertible. If the determinant is nonzero, then
the matrix is invertible. Next, we find the matrix of minors,
then the matrix of cofactors, then the adjoint matrix, and finally we multiply the
adjoint matrix by the reciprocal of the determinant to find the inverse matrix.
For the first step, we need to find
the determinant of 𝐴. We can do this by recalling the
formula for finding the determinant of a three-by-three matrix. Now, we can apply this formula to
find the determinant of 𝐴. Then, we simplify to find that the
determinant of 𝐴 is equal to negative three. Since this is nonzero, we know that
our matrix is invertible.
Next, we need to find the matrix of
minors . We find each entry of the matrix of minors by crossing out the row and
column that the entry is on and then finding the determinant of the remaining
two-by-two matrix, as shown here. We need to apply this to our matrix
𝐴. Following this method, we find our
matrix of minors. All we need to do now is evaluate
the determinants in the matrix. After doing this, we find that the
matrix of minors is negative three, six, zero, three, negative six, negative three,
negative two, three, one.
Now that we have found the matrix
of minors, we can move on to find the matrix of cofactors. When finding the matrix of
cofactors, we simply take the matrix of minors and change some of the signs. We change the signs of the entries
in the matrix of minors as shown to obtain the matrix of cofactors. We can apply this to the matrix of
minors we found in the last step to get our matrix of cofactors.
For the penultimate step, we need
to find the adjoint matrix. The adjoint matrix is simply the
transpose of the matrix of cofactors. So in order to find it, we simply
reflect the matrix of cofactors along its leading diagonal. Hence, the adjoint matrix is
negative three, negative three, negative two, negative six, negative six, negative
three, zero, three, one.
We are now ready to find the
inverse matrix. In order to do this, we simply
multiply the reciprocal of the determinant by the adjoint matrix.
We can now use this inverse matrix
to solve the system of linear equations. We can recall that we have 𝑋
equals the inverse of 𝐴 multiplied by 𝐵. To find the values of 𝑥, 𝑦, and
𝑧, we need to carry out the multiplication. We multiply each row of the matrix
by the column matrix and add together these products. Now, we simplify the column matrix
which we are left with. Now, all we have to do is multiply
our column matrix by the constant. We have that 𝑥, 𝑦, 𝑧 is equal to
negative one, zero, seven.
Hence, our solution to the system
of linear equations is 𝑥 equals negative one, 𝑦 equals zero, and 𝑧 equals