Answer:
![\left[\begin{array}{cccc}1&0&3&3\\0&1&0&0\\0&0&0&-1\\0&-2&1&1\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D1%260%263%263%5C%5C0%261%260%260%5C%5C0%260%260%26-1%5C%5C0%26-2%261%261%5Cend%7Barray%7D%5Cright%5D)
Step-by-step explanation:
Hi,
Considering that any elementary matrix can be obtained from the identity matrix of same dimensions using row operations, we consider our starting matrix to be:
![\left[\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D1%260%260%260%5C%5C0%261%260%260%5C%5C0%260%261%260%5C%5C0%260%260%261%5Cend%7Barray%7D%5Cright%5D)
For part a) we add the third row to fourth row, however nothing happens to the rest of the rows. Remember, the only change we see is in row 4.
![\left[\begin{array}{ccc}Row 1\\Row 2\\Row 3\\Row 4 + Row 3\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7DRow%201%5C%5CRow%202%5C%5CRow%203%5C%5CRow%204%20%2B%20Row%203%5Cend%7Barray%7D%5Cright%5D)
Addition in matrix is element-wise.
![\left[\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&1&0\\0+0&0+0&0+1&1+0\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D1%260%260%260%5C%5C0%261%260%260%5C%5C0%260%261%260%5C%5C0%2B0%260%2B0%260%2B1%261%2B0%5Cend%7Barray%7D%5Cright%5D)
We reach the following matrix at the end of part a)
![\left[\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&1&1\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D1%260%260%260%5C%5C0%261%260%260%5C%5C0%260%261%260%5C%5C0%260%261%261%5Cend%7Barray%7D%5Cright%5D)
b)
We begin with the matrix: ![\left[\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&1&1\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D1%260%260%260%5C%5C0%261%260%260%5C%5C0%260%261%260%5C%5C0%260%261%261%5Cend%7Barray%7D%5Cright%5D)
To subtract fourth row from third means:
![\left[\begin{array}{ccc}Row 1\\Row 2\\Row 3 - Row 4\\Row 4\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7DRow%201%5C%5CRow%202%5C%5CRow%203%20-%20Row%204%5C%5CRow%204%5Cend%7Barray%7D%5Cright%5D)
All matrix operations are element-wise:
![\left[\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0-0&0-0&1-1&0-1\\0&0&1&1\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D1%260%260%260%5C%5C0%261%260%260%5C%5C0-0%260-0%261-1%260-1%5C%5C0%260%261%261%5Cend%7Barray%7D%5Cright%5D)
we reach the following matrix at the end of part b)
![\left[\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&0&-1\\0&0&1&1\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D1%260%260%260%5C%5C0%261%260%260%5C%5C0%260%260%26-1%5C%5C0%260%261%261%5Cend%7Barray%7D%5Cright%5D)
c)
We shall continue this from the matrix we reached at end of part b)
Add 3 times Row 4 to Row 1:
![\left[\begin{array}{ccc}(3 times Row 4) + Row 1\\Row 2\\Row 3\\Row 4\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%283%20%20times%20Row%204%29%20%2B%20Row%201%5C%5CRow%202%5C%5CRow%203%5C%5CRow%204%5Cend%7Barray%7D%5Cright%5D)
Remember, all matrix operations are element-wise:
![\left[\begin{array}{cccc}(3*0) +1&(3*0)+0&(3*1)+0&(3*1)+0\\0&1&0&0\\0&0&0&-1\\0&0&1&1\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D%283%2A0%29%20%2B1%26%283%2A0%29%2B0%26%283%2A1%29%2B0%26%283%2A1%29%2B0%5C%5C0%261%260%260%5C%5C0%260%260%26-1%5C%5C0%260%261%261%5Cend%7Barray%7D%5Cright%5D)
Completing the operations in the matrix, we reach the following matrix at the end of part c)
![\left[\begin{array}{cccc}1&0&3&3\\0&1&0&0\\0&0&0&-1\\0&0&1&1\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D1%260%263%263%5C%5C0%261%260%260%5C%5C0%260%260%26-1%5C%5C0%260%261%261%5Cend%7Barray%7D%5Cright%5D)
d)
Continuing where we left in part c), we need to subtract two times the second row from the fourth row:
![\left[\begin{array}{ccc}Row 1\\Row 2\\Row 3\\Row 4 - (2 * Row 2)\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7DRow%201%5C%5CRow%202%5C%5CRow%203%5C%5CRow%204%20-%20%282%20%2A%20Row%202%29%5Cend%7Barray%7D%5Cright%5D)
Applying element-wise operations:
![\left[\begin{array}{cccc}1&0&3&3\\0&1&0&0\\0&0&0&-1\\0-(2*0)&0-(2*1)&1-(2*0)&1-(2*0)\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D1%260%263%263%5C%5C0%261%260%260%5C%5C0%260%260%26-1%5C%5C0-%282%2A0%29%260-%282%2A1%29%261-%282%2A0%29%261-%282%2A0%29%5Cend%7Barray%7D%5Cright%5D)
Completing the operations, we reach the following matrix at the end of part d)![\left[\begin{array}{cccc}1&0&3&3\\0&1&0&0\\0&0&0&-1\\0&-2&1&1\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D1%260%263%263%5C%5C0%261%260%260%5C%5C0%260%260%26-1%5C%5C0%26-2%261%261%5Cend%7Barray%7D%5Cright%5D)
This is the final answer after completing all operations on 4x4 matrix.