Originally Posted by

**s_ingram** Hi guys,

I am unable to get the answer to the following question: Find all the values of (x,y,z) which satisfy:

$\displaystyle \left( \begin{array}{ccc}

7 & 2 & 4 \\ 4 & 1 & 2 \\ 3 & 1 & 1 \end{array} \right)

\left( \begin{array}{c} x \\ y \\ z \end{array} \right) =

\left( \begin{array}{c} p \\ q \\ r \end{array} \right)

$

if (p,q,r) satisfies

$\displaystyle \left( \begin{array}{ccc}

2 & -7 & 5 \\ 6 & -9 & -1 \\ -4 & 5 & 2 \end{array} \right)

\left( \begin{array}{c} p \\ q \\ r \end{array} \right) =

\left( \begin{array}{c} 5 \\ 7 \\ -4 \end{array} \right)

$

So, we have:

$\displaystyle \mathbf{A}x = p$

detA = 7(-1) -2(-2) + 4(1) = 1. So it has an inverse. I used two methods to calculate the inverse, the adjoint and row reduction. Both gave the same result:

$\displaystyle A^{-1} = \left( \begin{array}{ccc}

-1 & 2 & 0 \\ -2 & -5 & 1 \\ 0 & -1 & -1 \end{array} \right)

$

which gives:

$\displaystyle x = 2q - p$

$\displaystyle y = 2p - 5q + 2r$

$\displaystyle z = p - q - r$

The second matrix B is singular i.e.

$\displaystyle det B = 2(-13) + 7(8) + 5(-6) = 0$ But we have the following relationships:

$\displaystyle 2p - 7q + 5r = 5$

$\displaystyle 6p - 9q - r = 7$

$\displaystyle -4p + 5q + 2r = -4$

I messed around and got:

p = 7x + 2y + 4z

q = 4x + y + 2z

r = 3x + y + z

but this doesn't help. The answers are given in terms of $\displaystyle \lambda!$ Am I missing something here?