6. Use Gaussian elimination to find the row echelon form of the matrices below. Show your work. What is the rank of each matrix? What is the dimension of its null space? Is the matrix invertible? For each non-invertible matrix, change one element to make it invertible. a. b. C. d. [512] 4 5 9 [12 3 32 1 5 2 6 1 -2 -7 1 9 2 782 "I do os 553 -1 9 -3 0 6 6 -

Note: "As per our guidelines we will solve the three subparts. If you want any specific subpart to be solved please specify that subpart or post only that subpart ."

(a).     $A= \left[\begin{array}{ccc}5& 1& 2\\ 4& 5& 6\\ 2& 9& 1\end{array}\right]$    

(b).    $B= \left[\begin{array}{ccc}1& 2& 3\\ 3& 2& 1\\ 5& 2& -1\end{array}\right]$

(c).    $C=\left[\begin{array}{ccc}-2& -7& 1\\ 6& 5& 9\\ 2& 3& 2\end{array} \right]$

(.)  Rank - nullity theorem :  Let $A$ be a             matrix of order $m\times n$ , then 

            $rank\left(A\right) + nullity\left(A\right) = n$                     i.e.  $n\to$number of columns .

=> Dimension of null space $=$ nullity 

(.)  Number of non - zero rows in row echelon form  is called  $\text{'}rank\text{'}$ .

(.)  If nullity of a matrix is  $zero$  then  the matrix is invertible .

(.)  Row Echelon form of a matrix :                                               A matrix is said to be in  row - echelon form if ,

(i)  All zero rows , if any are at the bottom          of the matrix .                                              (ii)  First non - zero element of every row is        on the right hand side of the first non -          zero element in the preceding row .

Example:-   $\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 2\\ 0& 0& 1\end{array}\right]$   $, \left[\begin{array}{ccc}2& 1& 1\\ 0& 1& 1\\ 0& 0& 0\end{array}\right]$

both matrix are in row - echelon form .

(.)   In Gaussian elimination method , we reduce the matrix in row echelon form by applying row operations .