Unless the matrix is square you won't be able to handle that, as arrays are immutable in size.

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| 1 2 3 |
| 4 5 6 |
== [[1,2,3],[4,5,6]]

| 1 2 3 | | 4 5 6 | == [[1,2,3],[4,5,6]]

Can't be put in the same array as

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| 1 4 |
| 2 5 |
| 3 6 |
== [[1,4],[2,5],[3,6]]

| 1 4 | | 2 5 | | 3 6 | == [[1,4],[2,5],[3,6]]

You're going to need some temporary storage anyway if you could get away with it (which you can with square matrices).
dest[i,j] = org[j,i] For each i,j;
Therefore you're going to have to store the value you're replacing until you've a place to put it (which is easy as you can quite readily figure out as you're effectively swapping valued in that scenario).

I think it is possible to represent a matrix by a linear array

If by linear u mean a 1 dimensional array, then yes thats possible. But to do so would B just stupid...

no, you don't save a byte using a one dimensional array instead of a multidimensional array.

And that's without the added address space needed for the accessfunctions (and the slower access to the data using those functions instead of whatever optimisations the JVM may employ on arrays).