GATE CSE 2017 | Set 2 | Question: 22

Question

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 ___________ .

Comments

Rank(A)+Rank(B) ≠ Rank(A+B)

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since once row LD therfore rank can not be three

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since once row LD therfore rank can not be three 
as we can see the R3 is linear combination of R1 and R2

Answer 1

$P +Q = \begin{bmatrix}0 & -1 &-2 \\8 & 9 & 10 \\8 & 8 & 8\end{bmatrix}$

$\det(P+Q) = 0$, So Rank cannot be $3$, but there exists a $2*2$ submatrix such that determinant of submatrix is not $0$.

So, $\text{Rank}(P+Q) = 2$

Comments

@Nandhakumar if Rank is Full then  eigen Value can't be $0$ and if eigen value is $0$ then Rank must be less than $n$ {No of Cols}

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why cant it be like swap row 1 and 3 then r2=r2-1 and then after that r3=r3+r2

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@FRUSTRATED_Engineer

R2 = R2-1 is not a valid elementary operation.

Answer 2

$P +Q = \begin{bmatrix}0 & -1 &-2 \\8 & 9 & 10 \\8 & 8 & 8\end{bmatrix}$

Applying $R_2 \leftarrow R_2+R_1$, we get

$P +Q = \begin{bmatrix}0 & -1 &-2 \\8 & 8 & 8 \\8 & 8 & 8\end{bmatrix}$

Since there are only $2$ independent rows $\implies Rank(P+Q) = 2$

Comments

Which are the pivot elements?

 

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@deep_184 it doesn't matter in method. 🤷‍♂️
But to answer your question check this: https://gateoverflow.in/118363/gate-cse-2017-set-2-question-22?show=430273#a430273

Answer 3

Rank(p+q) = 2

Comments

why cant it be like swap row 1 and 3 then r2=r2-1 and then after that r3=r3+r2

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@FRUSTRATED_Engineer Your wish brother
 

Answer 4

Apply these two row elementry operations on matrix Q

 

1. R3 -> 6R3 - 5R2

2. R2 -> R2 + 6R1

it will make 2 rows of Q zero. (#rank of Q=1)

now in echlon form of P we have two non zero rows(#rank of P=2) and if we also add Q it will not affect the rank(since we are not adding any nonzero element in last row of P).

Answer 5

Correct answer is 2

Answer 6

(P+Q) rank is 2