State and perform the next elementary row operations that should be performed to put the matrix in diagonal form

1 2 -3 3
2 -4 6 8
3 6 15 12

well, in that first column I would like to see

1
0
0

so I would multiply row 1 by 2
subtract the result from row two
and put that result where row 2 is

then I would multiply row 1 by 3 instead of by 2 and subtract the result from row three, placing the result in row three.

Damon,

Thank you for your help! Your explanation really helped me to understand the problem.

Would you be willing to help me with my question?

Yes, thank you Damon for your help with this. Hope you had a great 4th of July.

Krystal

To put the given matrix in diagonal form, you need to perform elementary row operations. Here are the steps you can follow:

Step 1: Swap rows if necessary
Check the first column (column 1) and see if any of the elements below the first row are non-zero. In this case, the second row has a non-zero element in the first column. Therefore, perform a row swap by exchanging the first row and the second row:

2 -4 6 8
1 2 -3 3
3 6 15 12

Step 2: Use row replacement to create zeros below the first element in column 1
To create zeros below the first element (2) in column 1, use row replacement. Multiply the first row by -1 and add it to the second row, and then multiply the first row by -3 and add it to the third row:

2 -4 6 8
0 6 -9 -13
0 18 -21 -12

Step 3: Use row replacement to create zeros below the second element in column 2
To create zeros below the second element (6) in column 2, use row replacement. Multiply the second row by 3 and add it to the third row:

2 -4 6 8
0 6 -9 -13
0 0 0 -61

Step 4: Swap rows as necessary
If there are any rows that have all zeros, you can move them to the bottom of the matrix. In this case, the third row has all zeros, so we can swap it with the last row:

2 -4 6 8
0 6 -9 -13
0 0 0 -61

Step 5: Divide rows by the leading coefficient
To put the matrix in diagonal form, divide each row by its leading coefficient (the coefficient of the first non-zero element in each row):

1 -2 3 4
0 1 -3/2 -13/6
0 0 0 1

Now, the matrix is in diagonal form. The leading coefficients of each row are 1, 1, and 1, which means that the matrix is diagonal.