Answer
$\left[\begin{array}{c}R \\ G \\ B\end{array}\right]=\left[\begin{array}{ccc}2.2586 & -1.0395 & -.3473 \\ -1.3495 & 2.3441 & .0696 \\ .0910 & -.3046 & 1.2777\end{array}\right]\left[\begin{array}{l}X \\ Y \\ Z\end{array}\right]$
Work Step by Step
Every matrix equation of the type $A X=Y$ can be tranformed to $X=$ $A^{-1} Y$
As:
$\left[\begin{array}{l}R \\ G \\ B\end{array}\right]=\left[\begin{array}{lll}.61 & .29 & .15 \\ .35 & .59 & .063 \\ .04 & .12 & .787\end{array}\right]^{-1}\left[\begin{array}{l}X \\ Y \\ Z\end{array}\right]$
Input matrix $A=\left[\begin{array}{ccc}.61 & .29 & .15 \\ .35 & .59 & .063 \\ .04 & .12 & .787\end{array}\right]$ in Matlab and use command inv $(A)$
to get its inverse.
Result is:
\[
\left[\begin{array}{c}
R \\
G \\
B
\end{array}\right]=\left[\begin{array}{ccc}
2.2586 & -1.0395 & -.3473 \\
-1.3495 & 2.3441 & .0696 \\
.0910 & -.3046 & 1.2777
\end{array}\right]\left[\begin{array}{c}
X \\
Y \\
Z
\end{array}\right]
\]
