When a deep space probe is launched, corrections may be necessary to place the probe on a precisely calculated trajectory. Radio telemetry provides a stream of vectors, $x_1,$dots,$x_k,$ giving information at different times about how the probe's position compares with its planned trajectory. Let $x_k$ be the matrix $].$ The matrix $G_k=x_k{x}_k^T$ is computed as the radar data are analyzed. When $x_{k+1}$ arrives, a new $G_{k+1}$ must be computed. Since the data vectors arrive at high speed, the computational burden could be severe. But partitioned matrix multiplication helps tremendously. Compute the column$-$row expansions of $G_k$ and $G_{k+1},$ and describe what must be computed in order to update $G_k$ to form $G_{k+1}.$

The probe Galileo was launched October $18,1989,$ and arrived near Jupiter in early December $1995.$