# According to Buckingham Theorem the rank of $A$ should be $2$

A physical system is described by a law of the form $f(E,P,A)=0$, where $E,P,A$ represent, respectively, energy, pressure and surface area. Find an equivalent physical law that relates suitable dimensionless quantities.

That' what I have tried so far:

1st step:

Choice of quantities

Mass: $M$

Time: $T$

Length: $L$

So:

$$[E]=M L^2 T^{-2}$$ $$[P]=ML^{-1}T^{-2}$$ $$[A]=L^2$$

2nd step:

Construction of dimonsionless quantities

The matrix of dimensions:

\begin{equation*} A=\begin{bmatrix} 1 & 1 & 0\\ -2 & -2 & 0 \\ 2 & -1 &2 \end{bmatrix} \end{equation*}

I tried to find the rank, determining the smallest $n$ for which $A^n=I$.

\begin{equation*} \begin{bmatrix} 1 & 1 & 0\\ -2 & -2 & 0 \\ 2 & -1 &2 \end{bmatrix}\begin{bmatrix} 1 & 1 & 0\\ -2 & -2 & 0 \\ 2 & -1 &2 \end{bmatrix}=\begin{bmatrix} -1 & -1 & 0\\ 2 & 2 & 0 \\ 8 & 2 &4 \end{bmatrix} \end{equation*}

\begin{equation*} \begin{bmatrix} -1 & -1 & 0\\ 2 & 2 & 0 \\ 8 & 2 &4 \end{bmatrix}\begin{bmatrix} 1 & 1 & 0\\ -2 & -2 & 0 \\ 2 & -1 &2 \end{bmatrix}=\begin{bmatrix} 1 & 1 & 0\\ -2 & -2 & 0 \\ 12 & 0 &8 \end{bmatrix} \end{equation*}

But I saw the solution and there should be only one dimensionless quantity, so according to Buckingham Theorem the rank of $A$ should be $2$.

Where is my mistake?

You don't need to do $A^n$.

You can see the 1st and 2nd row are multiple of each other, so one of them can be eliminated to $0$. That makes the rank $2$.

Alternatively, switch the 1st row and last row to make it

$$\pmatrix{2&-1&2\\-2&-2&0\\1&1&0}$$

Then divide the second row by $-2$, and subtract from the last row the resulting 2nd row. The result will be

$$\pmatrix{2&-1&2\\1&1&0\\0&0&0}$$

Or, you can do Gaussian elimination. First add $2$ times the 1st row to the second row, then subtract from 3rd row $2$ times the 1st row:

$$\pmatrix{1&1&0\\0&0&0\\0&-3&2}$$

Switching the last two rows:

$$\pmatrix{1&1&0\\0&-3&2\\0&0&0}$$

• So can we count the number of rows/ columns that are multiple of each other and this number will be equal to the rank of the matrix? Or elsewhise having done some operations or Gaussian elimination, do we count how many rows/ columns there are that contain only $0$s? Or have I understood it wrong? – evinda May 5 '15 at 20:36
• The safe way is to do Gaussian Elimination and count the number of nonzero rows after you reduce it to row echelon form. Counting number of multiples does not work if the matrix is complicated. – KittyL May 5 '15 at 22:53
• I see... Thanks a lot!!! :-) Could I also ask you something else? I am looking at the following exercise: – evinda May 5 '15 at 23:36
• A thin cover with the shape of a rectangle with mass per unit of volume equal to $m_f$ is put over a quantity of explosive ( with mass per unit of volume equal to $m_e$), that is attached at a base of a practically unbounded mass. – evinda May 5 '15 at 23:36
• If the explosive explodes, the cover is getting thrown vertically with velocity $v_f$. If $E_g$ is the so called "energy Gurney" of the explosive (in unites Joules/kg), i.e. the energy that the explosive material disposes to produce work, determine, as accurate as possible, using techniques of dimensional analysis, the velocity of the cover as a fuction of $m_f, m_e$ and $E_g$. (The exact relation that holds is the following: $v_f=\sqrt{2E_g} \left(\frac{m_f}{m_e}+\frac{1}{3} \right)^{-\frac{1}{2}}$)  Could you help me to find the quantities that we will use? – evinda May 5 '15 at 23:36