(A) The rank of a matrix is defined as the order of the largest non-singular sub-matrix.Since all sub-matrices of order $k$ are singular,the rank $\rh

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(A) The rank of a matrix is defined as the order of the largest non-singular sub-matrix.Since all sub-matrices of order $k$ are singular,the rank $\rho$ must be less than $k$,i.e.,$\rho Since there exists at least one non-singular sub-matrix of order $r$,the rank $\rho$ must be at least $r$,i.e.,$r \leq \rho$ ... (ii).Combining inequalities $(i)$ and (ii),we get $r \leq \rho < k$.

Class 12 Mathematics 3 and 4 .Determinants and Matrices Rank of Matrices , Some special determinants, differentiation and integration of determinants

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In a matrix $A$,if all the sub-matrices of order $k$ are singular and there is at least one non-singular sub-matrix of order $r$ $(r < k)$,then the rank $(\rho)$ of the matrix $A$:

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A
satisfies $r \leq \rho < k$
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is equal to $r$
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is equal to $(k-1)$
D
is equal to $(k+1)$

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3 and 4 .Determinants and Matrices

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Rank of Matrices , Some special determinants, differentiation and integration of determinants

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In a matrix $A$,if all the sub-matrices of order $k$ are singular and there is at least one non-singular sub-matrix of order $r$ $(r < k)$,then the rank $(\rho)$ of the matrix $A$:

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Let $f(x) = \left|\begin{array}{ccc} a & -1 & 0 \\ ax & a & -1 \\ ax^2 & ax & a \end{array}\right|$,where $a \in R$. Then the sum of the squares of all the values of $a$ for which $2f'(10) - f'(5) + 100 = 0$ is:

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