An apparatus and method for efficient matrix alignment in a systolic array. For example, one embodiment of a processor comprises: a first set of physical tile registers to store first matrix data in rows or columns; a second set of physical tile registers to store second matrix data in rows or columns; a decoder to decode a matrix instruction identifying a first input matrix, a first offset, a second input matrix, and a second offset; and execution circuitry, responsive to the matrix instruction, to read a subset of rows or columns from the first set of physical tile registers in accordance with the first offset, spanning multiple physical tile registers from the first set if indicated by the first offset to generate a first input matrix and the execution circuitry to read a subset of rows or columns from the second set of physical tile registers in accordance with the second offset, spanning multiple physical tile registers from the second set if indicated by the second offset to generate a second input matrix; and the execution circuitry to perform an arithmetic operation with the first and second input matrices in accordance with an opcode of the matrix instruction.