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I have this matrix A:

| 1 0 0 0 |
| 1 1 0 0 |
| 1 2 1 0 |
| 2 3 2 1 |

If I multiply the first row with (-1), and add it up to the second row, we have the matrix B:

| 1 0 0 0 |
| 0 1 0 0 |
| 1 2 1 0 |
| 2 3 2 1 |

And E is an unknown matrix where:

E*A = B

What is the value of the matrix E?

Basically, I'm trying to figure out the matrix which is equivalent to a computation in a matrix. I know that an option is to compute the inverse of A. However, I'm looking for a solution without inverse (as it's in the original problem I provided in the link below).

Originally this is the problem 2.2 from Algebra MIT OCW 18.06SC but simplified, as I'm struggling with this precise step.

Chris
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    Perform the same exact row operation on $I_{4\times 4}$ to get $E$ – Matthew H. Feb 01 '21 at 02:01
  • Each elementary row operation is represented by an elementary matrix. Have a look, see if you can figure out what $E$ is in this case. If you want to find the matrix representing multiple row operations, you can write out each elementary matrix and multiply all of them, or just do what Matthew suggests and perform the row operations on $I$. – Kyan Cheung Feb 01 '21 at 02:25

1 Answers1

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Try to convince yourself of the following by looking at the matrix multiplication formaula:

Let $E, A$ and $B$ be $n \times n$ matrices such that $ E A =B$. Let $\{E_{ik}\}_{k=1}^n$ the $i$th row elements of $E$ and let $A_k$ denote the rows of $A.$ (Note that $A_k$ is a $1\times n$ matrix). Then the $k$th row of B is given by $$ \sum_{k=1}^n E_{ik}A_k.$$ In your example:

$\begin{pmatrix} 1&&&\\ -1&1&&\\ &&1\\ &&&1\\ \end{pmatrix} \begin{pmatrix} 1& 0&0&0 \\ 1 &1 &0& 0 \\ 1 &2 &1& 0\\ 2 &3&2&1\ \end{pmatrix} =\begin{pmatrix} 1 & 0&0&0 \\ 0 &1 &0& 0 \\ 1 &2 &1& 0\\ 2 &3&2&1 \end{pmatrix}$

The second row of $B$ is given by minus one times first row of $E$ plus one times the second row of $E$ plus $0$ times the third row etc.

Asem Abdelraouf
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