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Q)

Find the rank of the following matrix :$\begin{bmatrix} 1 & 1 & -1 \\3 & -2 & 3 \\2 & -3 & 4 \end{bmatrix}$

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A)
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  • The rank of a matrix A is equal to $r$ if (i) A has atleast one minor of order r that does not vanish (ii) every minor of order $r+1$ or higher vanishes. $ \rho (A)=r.$
$ A = \begin{bmatrix}1 & 1 & -1 \\ 3 & -2 & 3 \\ 2 & -3 & 4 \end{bmatrix}$ is a 3 x 3 matrix.
$ \therefore \rho(A) \leq 3$
$ |A| = \begin{vmatrix}1 & 1 & -1 \\ 3 & -2 & 3 \\ 2 & -3 & 4 \end{vmatrix} = 1(-8+9)-1(12-6)+1(-9+4)$
$ = 1-6-5=10 \neq 0$
$ \therefore \rho(A) = 3$

 

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