Formulate the method for adding partitioned matrices, and verify your method by partitioning the matrices in two different ways and finding their sum. A=begin{bmatrix}1 & 3 & -1 2 & 1 & 0 2 & -3 &1 end{bmatrix} text{ and } B=begin{bmatrix}3 & 2 & 1 -2 & 3 & 1 4 & 1 &5 end{bmatrix}

Formulate the method for adding partitioned matrices,
and verify your method by partitioning the matrices in two different ways and finding their sum.
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Step 1
The given two matrices are ,
$A=\left[\begin{array}{ccc}1& 3& -1\\ 2& 1& 0\\ 2& -3& 1\end{array}\right]$
$B=\left[\begin{array}{ccc}3& 2& 1\\ -2& 3& 1\\ 4& 1& 5\end{array}\right]$
finding the sum of matrices , A+B
$A+B=\left[\begin{array}{ccc}1& 3& -1\\ 2& 1& 0\\ 2& -3& 1\end{array}\right]+\left[\begin{array}{ccc}3& 2& 1\\ -2& 3& 1\\ 4& 1& 5\end{array}\right]$
$=\left[\begin{array}{ccc}\left(1+3\right)& \left(3+2\right)& \left(-1+1\right)\\ \left(2+\left(-2\right)\right)& \left(1+3\right)& \left(0+1\right)\\ \left(2+4\right)& \left(-3+1\right)& \left(1+5\right)\end{array}\right]$
$=\left[\begin{array}{ccc}4& 5& 0\\ 0& 4& 1\\ 6& -2& 6\end{array}\right]$
Therefore , the sum of the given two matrices is $\left[\begin{array}{ccc}4& 5& 0\\ 0& 4& 1\\ 6& -2& 6\end{array}\right]$
Step 2
To verify it ,we change the position of the matrices and find the sum of B+A
Finding the sum of matrices , B+A
$B+A=\left[\begin{array}{ccc}3& 2& 1\\ -2& 3& 1\\ 4& 1& 5\end{array}\right]+\left[\begin{array}{ccc}1& 3& -1\\ 2& 1& 0\\ 2& -3& 1\end{array}\right]$
$=\left[\begin{array}{ccc}\left(3+1\right)& \left(2+3\right)& \left(1+\left(-1\right)\right)\\ \left(\left(-2\right)+2\right)& \left(3+1\right)& \left(1+0\right)\\ \left(4+2\right)& \left(1+\left(-3\right)\right)& \left(5+1\right)\end{array}\right]$
$=\left[\begin{array}{ccc}4& 5& 0\\ 0& 4& 1\\ 6& -2& 6\end{array}\right]$
Therefore , the sum of the given two matrices is $\left[\begin{array}{ccc}4& 5& 0\\ 0& 4& 1\\ 6& -2& 6\end{array}\right]$
Jeffrey Jordon