Rank of the matrix

Question

Given that a matrix $[A]_{4\times4},$any one row/column is dependent on the others, and given matrix are singular matrix$(|A|=0)$.

And another matrix $B=adj(A),$then find them,

$1)$Rank of the matrix $B$

$2)$Rank of the marix $adj(B)$

Comments

https://math.stackexchange.com/questions/260298/matrix-a-is-regular-iff-its-adjugate-matrix-is-regular/260326#260326

https://gateoverflow.in/9805/rank

Answer 1

let take a 3x3 matrix instead of 4x4 matrix for simplicity

The rank of a matrix is defined as the maximum number of linearly independent row/ col vectors in the matrix

Here it's given that  " one row/column is dependent on the others"

which means out of 3 rows , let R1 , R2 rows are linearly independent and  R3 is linear dependent

therefore , rank of the matrix B  = 2

therefore rank of (A) = 2

Now if A is singular then adj(A) is also a singular

 

A = $\begin{bmatrix} a & b & c & \\ d & e&f & \\ 0&0 & 0 & \end{bmatrix}$

B = adj(A) = 

      $\begin{bmatrix} 0 & 0 & 0 & \\ 0 & 0&0 & \\ bf -e c&dc-af & ae -db & \end{bmatrix}$

 

 

     $\begin{bmatrix} bf -e c & dc-af & ae -db& \\ 0 & 0&0 & \\ 0 & 0& 0 & \end{bmatrix}$ = rank  = 1

 

Now adj(B) is a null matrix

therefore rank = 0

Comments

for 2x2 matrix you can't get the null matrix

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what is wrong to take $2\times2$

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what is wrong to take 2×2

Rank of A=3

1)Rank of B=1

2)Rank of adj(B)=0

you couldn't get this 3 conditions