<s>[INST] Find unknown coefficient of a matrix with known rank. Let there be a matrix A = $\begin{bmatrix} 8 & 7 & 5 \\ 4 & 5 & 6 \\ 7 & 8 & λ \end{bmatrix}$.
Find the λ value, for which matrix A has a rank(A) = 2.
First I need to do the reduced row echelon form (I guess). Should I start by dividing the first row by 8 or by substracting the third row from the first? Will the result be the same? How should I proceed with the information rank(A) = 2 given?
 [/INST] Hint:  You need row $3$ to be a linear combination of rows $1$ and $2$.  Write $a(8,7,5)+b(4,5,6)=(7,8,\lambda) $.
Solve the system  $\begin{cases}{8a+4b=7 \\7a+5b=8}\end{cases}$ for $a$ and $b$.
Then $\lambda =5a+6b$.
If we let $M=\begin{pmatrix}8&4\\7&5\end{pmatrix}$, then $M^{-1}=\frac1{12}\begin{pmatrix}5&-4\\-7&8\end{pmatrix}$.
Thus 

$\begin{pmatrix} a\\b\end{pmatrix}=M^{-1}\begin{pmatrix}7\\8\end{pmatrix}=\begin{pmatrix}\frac14\\\frac54\end{pmatrix}\,\therefore \lambda =\frac{35}4$

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