# Thread: [SOLVED] Solving this matrix for C

1. ## [SOLVED] Solving this matrix for C

Hi I am trying to solve NC=E to find C. I was thinking I need the inverse of N but before I do that I wanted to check that there is not an easier way to solve for C?

Matrices are:

5x2 matrix
C=
R1(c11,c12)
R2(c21,c22)
R3(c31,c32)
R4(c41,c42)
R5(c51,c52)

5x2 matrix
E=
R1(43,7)
R2(41,4)
R3(29,5)
R4(57,10)
R5(20,6)

5X5 Matrix
N=
R1(1,2,2,3,3)
R2(1,1,3,2,2)
R3(2,1,2,1,2)
R4(1,3,3,2,1)
R5(3,1,1,1,3)

2. Originally Posted by Neverquit
Hi I am trying to solve NC=E to find C. I was thinking I need the inverse of N but before I do that I wanted to check that there is not an easier way to solve for C?

Matrices are:

5x2 matrix
C=
R1(c11,c12)
R2(c21,c22)
R3(c31,c32)
R4(c41,c42)
R5(c51,c52)

5x2 matrix
E=
R1(43,7)
R2(41,4)
R3(29,5)
R4(57,10)
R5(20,6)

5X5 Matrix
N=
R1(1,2,2,3,3)
R2(1,1,3,2,2)
R3(2,1,2,1,2)
R4(1,3,3,2,1)
R5(3,1,1,1,3)
$\displaystyle \left[\begin{matrix}1 & 2 & 2 & 3 & 3 \\ 1 & 1 & 3 & 2 & 2 \\ 2 & 1 & 2 & 1 & 2 \\ 1 & 3 & 3 & 2 & 1 \\ 3 & 1 & 1 & 1 & 3\end{matrix}\right]\left[\begin{matrix}c_{11} & c_{12} \\ c_{21} & c_{22}\\ c_{31} & c_{32} \\ c_{41} & c_{42} \\ c_{51} & c_{52}\end{matrix}\right] = \left[\begin{matrix}43 & 7 \\ 41 & 4 \\ 29 & 5\\ 57 & 10 \\ 20 & 6 \end{matrix}\right]$
You can premultiply both sides by $\displaystyle N^{-1}$, or you can do row operations.
Apply $\displaystyle R_2 - R_1 \to R_2, R_3 - 2R_1 \to R_3, R_4 - R_1 \to R_4, R_5 - 3R_1 \to R_1$
$\displaystyle \left[\begin{matrix}\phantom{-}1 & \phantom{-}2 & \phantom{-}2 & \phantom{-}3 & \phantom{-}3 \\ \phantom{-}0 & -1 & \phantom{-}1 & -1 & -1 \\ \phantom{-}0 & -3 & -2 & -5 & -4 \\ \phantom{-}0 & \phantom{-}1 & \phantom{-}1 & -1 & -2 \\ \phantom{-}0 & -5 & -5 & -8 & -6\end{matrix}\right]\left[\begin{matrix}c_{11} & c_{12} \\ c_{21} & c_{22}\\ c_{31} & c_{32} \\ c_{41} & c_{42} \\ c_{51} & c_{52}\end{matrix}\right]$$\displaystyle =\left[\begin{matrix}43 & \phantom{-}7 \\ -2 & -3 \\ -57 & -9 \\ \phantom{-}14 & \phantom{-}3 \\ -106 & -13\end{matrix}\right]$
Go from here to try and upper triangularise the system. Then you can evaluate $\displaystyle C$ by solving and back substituting.