Let $P = \begin{bmatrix} 1 & 1 & -1 \\ 2 & -3 & 4 \\ 3 & -2 & 3 \end{bmatrix}$ and $Q = \begin{bmatrix} -1 & -2 & -1 \\ 6 & 12 & 6 \\ 5 & 10 & 5 \end{bmatrix}$ be two matrices. Then the rank of $P + Q$ is _________ .

First, compute the matrix sum $P + Q$:

Let $R_1, R_2, R_3$ denote the rows of $P + Q$:

• $R_1 = \begin{bmatrix} 0 & -1 & -2 \end{bmatrix}$

• $R_2 = \begin{bmatrix} 8 & 9 & 10 \end{bmatrix}$

• $R_3 = \begin{bmatrix} 8 & 8 & 8 \end{bmatrix}$

Notice the linear relation among the rows:

Thus, $R_2 = R_3 - R_1$, showing that $R_2$ is a linear combination of $R_1$ and $R_3$.

Since $R_1$ and $R_3$ are not scalar multiples of each other, they are linearly independent. Hence, the number of linearly independent rows is $2$.

Therefore, the rank of $P + Q$ is 2.