GO Classes DPP | GATE CS, DA | Linear Algebra | Rank-Nullity Theormem

Answer 1

Since $A$ is a $3\times 3$ matrix and nullity$(A)=1$, by rank-nullity theorem,

$\operatorname{rank}(A)+\operatorname{nullity}(A)=3$

$\operatorname{rank}(A)+1=3$

$\operatorname{rank}(A)=2$

So $\det(A)=0$.

Now,

$\det(A)=\begin{vmatrix}2&3&7\\1&1&9\\9&2&x\end{vmatrix}$

$=2\begin{vmatrix}1&9\\2&x\end{vmatrix}-3\begin{vmatrix}1&9\\9&x\end{vmatrix}+7\begin{vmatrix}1&1\\9&2\end{vmatrix}$

$=2(x-18)-3(x-81)+7(2-9)$

$=2x-36-3x+243-49$

$=158-x$

Since $\det(A)=0$,

$158-x=0$

$x=158$

Hence, $\boxed{x=158}$.

Answer 2

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Answer 3

Ans : 158 , calculating rank of  matrices by row echlon  from .

Answer 4

For a 3*3 matrix, the rank nullity theorem states that :

Rank+ Nullity=3
Find the rank :Since nullity is 1 , the rank must be 2 (3-1==2).
Linear Dependency: A rank of 2  (which is less than the matrix size of 3 ) means the rows or columns are linearly dependent.

The Determinant: Because the rows are dependent, the matrix is singular, meaning det(A)=0.

Solve for x : We can find the value of x using det(A)=0.