Answer
\[
\left[\begin{array}{c}
R \\
G \\
B
\end{array}\right]=\left[\begin{array}{ccc}
1.0031 & .9548 & .6179 \\
.9968 & -.2707 & -.6448 \\
1.0085 & -1.1105 & 1.6996
\end{array}\right]\left[\begin{array}{c}
Y \\
I \\
Q
\end{array}\right]
\]
Work Step by Step
Every matrix equation of the type $A X=Y$ can be tranformed to $X=$ $A^{-1} Y$
As:
\[
\left[\begin{array}{c}
R \\
G \\
B
\end{array}\right]=\left[\begin{array}{ccc}
.299 & .587 & .114 \\
.596 & -.275 & -.321 \\
.212 & -.528 & .311
\end{array}\right]^{-1}\left[\begin{array}{c}
Y \\
I \\
Q
\end{array}\right]
\]
By putting matrix $A=\left[\begin{array}{ccc}299 & .587 & .114 \\ .596 & -.275 & -.321 \\ .212 & -.528 & .311\end{array}\right]$ in Matlab and use command
$\operatorname{inv}(A)$ to find its inverse.
Result is:
\[
\left[\begin{array}{c}
R \\
G \\
B
\end{array}\right]=\left[\begin{array}{ccc}
1.0031 & .9548 & .6179 \\
.9968 & -.2707 & -.6448 \\
1.0085 & -1.1105 & 1.6996
\end{array}\right]\left[\begin{array}{c}
Y \\
I \\
Q
\end{array}\right]
\]
