Question

Change the following matrix into upper triangular.
`\left[\begin{array}{ccc}7&2&4\\1&3&2\\-1&1&6\end{array}\right]`

Answer 1

To change the given matrix to an upper triangular matrix, elementary row operations are applied to create zeros below the main diagonal, step by step, starting from the first row and moving downwards.

Step-by-step explanation:

To change the given matrix into an upper triangular matrix, we need to perform elementary row operations to create zeros below the main diagonal. The matrix we are starting with is:

[7 2 4]
[1 3 2]
[-1 1 6]

First, we can multiply the first row by 1/7 (to make the leading coefficient 1) and then use it to eliminate the terms below it.

Now the matrix looks like this:

[1 2/7 4/7]
[1 3 2]
[-1 1 6]

We subtract the first row from the second row and add the first row to the third one, obtaining:

[1 2/7 4/7]
[0 3-(2/7) 2-(4/7)]
[0 1+(2/7) 6+(4/7)]

Now we can multiply the second row by 7/20 (to get a leading coefficient of 1) and use it to eliminate the term below it.

With some calculations, the second and third rows are now:

[0 1 20/21]
[0 0 6+4/7-20/21]

Hence, the upper triangular matrix is:

[1 2/7 4/7]
[0 1 20/21]
[0 0 6+4/7-20/21]

We can simplify the final entry to obtain the fully simplified upper triangular matrix:

[1 2/7 4/7]
[0 1 20/21]
[0 0 126/7+4/7-20/21]

Note: We can simplify the entries further if required.