31 votes 31 votes 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 ___________ . Linear Algebra gatecse-2017-set2 linear-algebra eigen-value numerical-answers + – Arjun asked Feb 14, 2017 edited Feb 15, 2017 by mcjoshi Arjun 11.7k views answer comment Share Follow See all 2 Comments See all 2 2 Comments reply ayush.5 commented Oct 12, 2020 reply Follow Share 1st column of P+Q is linear combination of 2nd and 3rd column as col1=2*col2-col3. So rank cannot be 3 3 votes 3 votes Souvik33 commented Dec 7, 2022 reply Follow Share Anyone who tried to solve usingRank(A+B) = Rank(A)+Rank(B) and got answer as 3, the problem is it is not an equality equation.The equation is:Rank(A+B) ≤ Rank(A)+Rank(B) and 2 is clearly less than 3 2 votes 2 votes Please log in or register to add a comment.
Best answer 52 votes 52 votes $P +Q = \begin{bmatrix}0 & -1 &-2 \\8 & 9 & 10 \\8 & 8 & 8\end{bmatrix}$ $\det(P+Q) = 0$, So Rank cannot be $3$, but there exists a $2*2$ submatrix such that determinant of submatrix is not $0$. So, $\text{Rank}(P+Q) = 2$ mcjoshi answered Feb 14, 2017 edited Jun 19, 2021 by Lakshman Bhaiya mcjoshi comment Share Follow See all 9 Comments See all 9 9 Comments reply Show 6 previous comments shashankrustagi commented Dec 5, 2020 reply Follow Share 1st and 2nd or 3rd and 2nd 0 votes 0 votes abir_banerjee commented Oct 9, 2022 reply Follow Share Rank of matrix tells about a lot of things :- Number of linear independent rows . Number of linear independent columns. Number of non zero rows One thing to note is that number of linear independent rows are equal to number of linear independent columns. 2 votes 2 votes Nandhakumar commented Oct 12, 2023 reply Follow Share Rank also tells the number of non zero eigen values right? 0 votes 0 votes Please log in or register to add a comment.
7 votes 7 votes $P +Q = \begin{bmatrix}0 & -1 &-2 \\8 & 9 & 10 \\8 & 8 & 8\end{bmatrix}$ 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 $\implies Rank(P+Q) = 2$ Satbir answered Oct 23, 2019 Satbir comment Share Follow See all 0 reply Please log in or register to add a comment.