Element Wise Matrix Addition/Subtraction and NULL Matrix

1. Element Wise Matrix Addition/Subtraction between any number of Matrices can be done if all the Matrices have Same Number of Rows and Same Number of Columns. The Resultant Matrix also has the Same Number of Rows and Columns as the input Matrices.
2. Given 2 \(M \times N\) Matrix \(A\) having elements \(a_{ij}\) and Matrix \(B\) having elements \(b_{ij}\) as following
   
   \(A=\begin{bmatrix} a_{11} & a_{12} & ... & a_{1n}\\ a_{21} & a_{22} & ... & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & ... & a_{mn}\end{bmatrix} \hspace{.5cm}B=\begin{bmatrix} b_{11} & b_{12} & ... & b_{1n}\\ b_{21} & b_{22} & ... & b_{2n} \\ \vdots & \vdots & \ddots & \vdots \\b_{m1} & b_{m2} & ... & b_{mn}\end{bmatrix}\)
   
   Element Wise Matrix Addition/Subtraction between Matrix \(A\) and Matrix \(B\) is given as follows
   
   \(A \pm B=\begin{bmatrix} a_{11} & a_{12} & ... & a_{1n}\\ a_{21} & a_{22} & ... & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & ... & a_{mn}\end{bmatrix} \pm \begin{bmatrix} b_{11} & b_{12} & ... & b_{1n}\\ b_{21} & b_{22} & ... & b_{2n} \\ \vdots & \vdots & \ddots & \vdots \\b_{m1} & b_{m2} & ... & b_{mn}\end{bmatrix}= \begin{bmatrix} a_{11}\pm b_{11} & a_{12}\pm b_{12} & ... & a_{1n}\pm b_{1n}\\ a_{21}\pm b_{21} & a_{22}\pm b_{22} & ... & a_{2n}\pm b_{2n}\\ \vdots & \vdots & \ddots & \vdots \\a_{m1}\pm b_{m1} & a_{m2}\pm b_{m2} & ... & a_{mn}\pm b_{mn}\end{bmatrix}\)
   
   
3. Following are some examples of Element Wise Matrix Addition/Subtraction.
   
   \(\begin{bmatrix} 2 & 9 & 1\\ 3 & 1 & 2\end{bmatrix} + \begin{bmatrix} -3 & 7 & 2 \\ 6 & -5 & 3\end{bmatrix} = \begin{bmatrix} 2 + (-3) & 9 + 7 & 1 + 2\\ 3+6 & 1 + (-5) & 2 +3 \end{bmatrix} = \begin{bmatrix} -1 & 16 & 3\\ 9 & -4 & 5\end{bmatrix}\)
   
   \(\begin{bmatrix} 7 & 2 \\ -6 & 5 \\ 4 & -3 \end{bmatrix} - \begin{bmatrix} -2 & 9 \\ 3 & -13 \\ 11 & 1 \end{bmatrix} = \begin{bmatrix} 7-(-2) & 2-9\\ -6-3 & 5-(-13)\\ 4-11 & -3-1 \end{bmatrix} = \begin{bmatrix} 9 & -7\\ -9 & 18\\ -7 & -4 \end{bmatrix}\)
   
   
4. Any Matrix having all it's elements set to Numerical Value 0 is called a NULL Matrix. Following are some examples of NULL Matrices
   
   \(\begin{bmatrix} 0 & 0 & 0\\ 0 & 0 & 0\end{bmatrix}\hspace{.5cm} \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0\\0 & 0 & 0 \end{bmatrix}\hspace{.5cm} \begin{bmatrix} 0 & 0 \end{bmatrix}\hspace{.5cm} \begin{bmatrix} 0 \\ 0 \\0 \\0 \end{bmatrix}\)
   
   NULL Matrices can be/are also represented by a Numerical Value 0.
5. Adding any Matrix to a NULL Matrix or Subtracting a NULL Matrix from any Matrix gives back the Original Matrix. Subtracting a Matrix from a NULL Matrix gives Negative of the Matrix. For example, given a \(M \times N\) Matrix \(A\), following are the results of performing Addition/Subtraction operations on Matrix \(A\) with a \(M \times N\) NULL Matrix.
   
   \(A + 0 = A\)
   \(A - 0 = A\)
   \(0 - A = -A\)
6. You can use the Matrix Sum and Difference Calculator to Add and Subtract Matrices.