GATE CS 2017 Set 2 Q.22 (CS Q.22): Linear Algebra

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

Your answer

Official answer and solution

Official answer: 2
From: Official GATE 2017 answer key

Solution

Compute the sum matrix $P + Q$ and perform elementary row operations to determine the number of linearly independent rows.

Compute matrix $P + Q$ entry by entry:

P + Q = \begin{bmatrix} 1 & 1 & -1 \\ 2 & -3 & 4 \\ 3 & -2 & 3 \end{bmatrix} + \begin{bmatrix} -1 & -2 & -1 \\ 6 & 12 & 6 \\ 5 & 10 & 5 \end{bmatrix} = \begin{bmatrix} 0 & -1 & -2 \\ 8 & 9 & 10 \\ 8 & 8 & 8 \end{bmatrix}

Perform elementary row operations to find the row echelon form:

Subtract Row 3 from Row 2 ( $R_2 \to R_2 - R_3$ ):

R_2 - R_3 = [8-8,\ 9-8,\ 10-8] = [0,\ 1,\ 2]

Notice that Row 1 is $[0, -1, -2]$ .
Add Row 1 to the new Row 2 ( $R_1 \to R_1 + (R_2 - R_3)$ ):

[0, -1, -2] + [0, 1, 2] = [0, 0, 0]

Row 1 is an exact linear combination of Row 2 and Row 3: $R_1 = R_3 - R_2$ .

Rows 2 and 3 ( $[8, 9, 10]$ and $[8, 8, 8]$ ) are not scalar multiples of each other, so they are linearly independent.
Thus, the row echelon form of $P + Q$ has exactly 2 non-zero rows, meaning $\text{rank}(P + Q) = 2$ .

Final answer

Common trap

Computing $\text{rank}(P) + \text{rank}(Q)$ instead of $\text{rank}(P + Q)$ .

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2017 Computer Science and Information Technology (CS) Session 2
Question: 22 of 65, printed as CS Q.22
Section: Computer Science and Information Technology

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