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Answer:
To see the steps to the diagonal form see the step-by-step explanation. The solution to the system is ,
,
and
Step-by-step explanation:
Gauss elimination method consists in reducing the matrix to a upper triangular one by using three different types of row operations (this is why the method is also called row reduction method). The three elementary row operations are:
- Swapping two rows
- Multiplying a row by a nonzero number
- Adding a multiple of one row to another row
To solve the system using the Gauss elimination method we need to write the augmented matrix of the system. For the given system, this matrix is:
For this matrix we need to perform the following row operations:
(multiply 1 row by 1 and subtract it from 2 row)
(multiply 1 row by 1 and add it to 3 row)
(multiply 1 row by 1 and subtract it from 4 row)
(interchange the 2 and 3 rows)
(divide the 2 row by 2)
(multiply 2 row by 1 and subtract it from 1 row)
(multiply 2 row by 1 and subtract it from 4 row)
(multiply the 3 row by -1)
(multiply 3 row by 1 and subtract it from 2 row)
(multiply 3 row by 3 and add it to 4 row)
(divide the 4 row by 4.5)
After this step, the system has an upper triangular form
The triangular matrix looks like:
If you later perform the following operations you can find the solution to the system.
(multiply 4 row by 0.5 and add it to 1 row)
(multiply 4 row by 0.5 and add it to 2 row)
(multiply 4 row by 2 and subtract it from 3 row)
After this operations, the matrix should look like:
Thus, the solution is:
,
,
and
Looks like .
- If
, then
- critical point at (0, 0).
- If
, then
- two critical points at
and
The latter two critical points occur outside of since
so we ignore those points.
The Hessian matrix for this function is
The value of its determinant at (0, 0) is , which means a minimum occurs at the point, and we have
.
Now consider each boundary:
- If
, then
which has 3 extreme values over the interval of 31/4 = 7.75 at the point (1, 1/2); 8 at (1, 1); and 10 at (1, -1).
- If
, then
and we get the same extrema as in the previous case: 8 at (-1, 1), and 10 at (-1, -1).
- If
which doesn't tell us about anything we don't already know (namely that 8 is an extreme value).
- If
, then
which has 3 extreme values, but the previous cases already include them.
Hence has absolute maxima of 10 at the points (1, -1) and (-1, -1) and an absolute minimum of 0 at (0, 0).