What is an n × n matrix A(i, j, c) such that if $B \in M_{n×m}$, then A(i, j, c)B is the matrix obtained from B by replacing its $i^{th}$ row by the sum of the $i^{th}$ row and the {j^{th}$ row multiplied by c?
I am a little confused to what A(i, j, c) is exactly here, i know it is a square matrix but what is i, j, c. I am not sure how to tackle this to figure out what A(i, j, c) is like so any hints to help me get started would be great.
$A(i,j,c)$ is a matrix that is obtained from the identity $I$ by replacing by $c$ the zero in the $(i,j)^{th}$ position of $I.$ This will give you the desired effect. Here is an example: suppose
$A=\begin{pmatrix} a &b &c \\ e &f &g \\ h&k &l \end{pmatrix}.\quad $ Then $\quad A(3,1,-2)=\begin{pmatrix} 1 &0 &0 \\ 0 &1 &0 \\ -2&0 &1 \end{pmatrix}\quad $ and $A(3,1,-2)A=\begin{pmatrix} 1 &0 &0 \\ 0 &1 &0 \\ -2&0 &1 \end{pmatrix}\begin{pmatrix} a &b &c \\ e &f &g \\ h&k &l \end{pmatrix}=\begin{pmatrix} a &b &c \\ e &f &g \\ -2a+h&-2b+k &-2a+l \end{pmatrix}$
So, $A(3,1,-2)$ multiplies the first row of $A$ by $-2$, adds this to the third row and replaces the third row by the result.