Question

Write down the elementary matrix corresponding to the following row operations on 4 × 4 matrices: (a) Add the third row to the fourth row. (b) Subtract the fourth row from the third row. (c) Add 3 times the last row to the first row. (d) Subtract twice the second row from the fourth row.

Answer 1

Answer:

$\left[\begin{array}{cccc}1&0&3&3\\0&1&0&0\\0&0&0&-1\\0&-2&1&1\end{array}\right]$

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:

$\left[\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{array}\right]$

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.

$\left[\begin{array}{ccc}Row 1\\Row 2\\Row 3\\Row 4 + Row 3\end{array}\right]$

Addition in matrix is element-wise.

$\left[\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&1&0\\0+0&0+0&0+1&1+0\end{array}\right]$

We reach the following matrix at the end of part a)

$\left[\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&1&1\end{array}\right]$

b)

We begin with the matrix: $\left[\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&1&1\end{array}\right]$

To subtract fourth row from third means:

$\left[\begin{array}{ccc}Row 1\\Row 2\\Row 3 - Row 4\\Row 4\end{array}\right]$

All matrix operations are element-wise:

$\left[\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0-0&0-0&1-1&0-1\\0&0&1&1\end{array}\right]$

we reach the following matrix at the end of part b)

$\left[\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&0&-1\\0&0&1&1\end{array}\right]$

c)

We shall continue this from the matrix we reached at end of part b)

Add 3 times Row 4 to Row 1:

$\left[\begin{array}{ccc}(3 times Row 4) + Row 1\\Row 2\\Row 3\\Row 4\end{array}\right]$

Remember, all matrix operations are element-wise:

$\left[\begin{array}{cccc}(3*0) +1&(3*0)+0&(3*1)+0&(3*1)+0\\0&1&0&0\\0&0&0&-1\\0&0&1&1\end{array}\right]$

Completing the operations in the matrix, we reach the following matrix at the end of part c)

$\left[\begin{array}{cccc}1&0&3&3\\0&1&0&0\\0&0&0&-1\\0&0&1&1\end{array}\right]$

d)

Continuing where we left in part c), we need to subtract two times the second row from the fourth row:

$\left[\begin{array}{ccc}Row 1\\Row 2\\Row 3\\Row 4 - (2 * Row 2)\end{array}\right]$

Applying element-wise operations:

$\left[\begin{array}{cccc}1&0&3&3\\0&1&0&0\\0&0&0&-1\\0-(2*0)&0-(2*1)&1-(2*0)&1-(2*0)\end{array}\right]$

Completing the operations, we reach the following matrix at the end of part d)$\left[\begin{array}{cccc}1&0&3&3\\0&1&0&0\\0&0&0&-1\\0&-2&1&1\end{array}\right]$

This is the final answer after completing all operations on 4x4 matrix.