Definition of terms:
This conversion is better avoided, since it is nonlinier and has singularities at + &  90 degrees, so if already working in terms of matricies its better to continue using matrices if possible.
This depends on what conventions are used for the Euler Angles. The following assumes NASA Standard Airplane
So it we look at the Euler to Matrix conversion we can see that:
m01/m00 = s1 / c1 = tan(heading)
m12/m22 = s3 / c3 = tan(bank)
m02 = s2 = sin(attitude)
so this gives:
heading = atan(m01/m00)
bank = atan(m12/m22)
attitude = asin(m02)
Note this only applies to a martix which represents a pure rotation. The equations for heading and bank should be independent of uniform scaling as it will cancel out in the division. It would be better to find an expression for attitude which is also independent of scaling.
Since there are several ways to produce the same rotation using heading, bank and attitude then the solution is not unique. tan(0) is 0, tan(90 degrees) is infinity, tan(90 degrees) is infinity. So the results will depend on whether arctan processes a result between 90 and 90 or 0 and 180.
we take the 90 degree rotation from this:  to this: 
As shown here the matrix for this rotation is:
[R] = 

So using the above result:
heading = atan(m01/m00) = atan(0) = 0
bank = atan(m12/m22) = atan(infinity) = 90 degrees
attitude = asin(m02) = asin(0) = 0
As you can see here, this gives the result that we are looking for.
We have to be very careful about the following issues:
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see also: 

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