A Gaussian elimination computing system and a Gaussian elimination computing method are provided. The Gaussian elimination computing system includes a control circuit, a systolic array, and a memory. The control circuit receives an operation matrix. The systolic array includes a square array formed by a plurality of operating cells. The systolic array is configured to perform a matrix decomposition operation to the operation matrix, to decompose the operation matrix into a lower triangular matrix and an upper triangular matrix. The memory is configured with an operation data block with the same size as the operation matrix for storing the lower triangular matrix and the upper triangular matrix after decomposition.