We proceed to develop the algebra of addition of matrices.
Two matrices A and B are said to be conformable for addition if they have the same order (i.e., same number of rows and columns).
If A = (a_{ij})_{m, n} and B = (b_{ij})_{m, n} then their sum A + B is the matrix C = (c_{ij})_{m,n} where c_{ij }= a_{ij} + b_{ij}, i = 1, 2, 3, ...... , m, j = 1, 2, 3, ...., n.
For example:
If A = \(\begin{bmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{bmatrix}\) and B = \(\begin{bmatrix} b_{11} & b_{12} & b_{13}\\ b_{21} & b_{22} & b_{23}\\ b_{31} & b_{32} & b_{33} \end{bmatrix}\), then
A + B = \(\begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} & a_{13} + b_{13}\\ a_{21} + b_{21} & a_{22} + b_{22} & a_{23} + b_{23}\\ a_{31} + b_{31} & a_{32} + b_{32} & a_{33} + b_{33} \end{bmatrix}\) = C
Note: If A and B be matrices of different orders, then A + B is not defined.
Example on Addition of Matrices:
1. If A = \(\begin{bmatrix} 2 & 5\\ -1 & 3 \end{bmatrix}\) and B = \(\begin{bmatrix} 1 & 4\\ 3 & 7 \end{bmatrix}\), then
A + B = \(\begin{bmatrix} 2 + 1 & 5 + 4\\ -1 + 3 & 3 + 7\end{bmatrix}\)
= \(\begin{bmatrix} 3 & 9\\ 2 & 10 \end{bmatrix}\)
2. If A = \(\begin{bmatrix} -1 & 2 & 3\\ 2 & -3 & 1\\ 3 & 1 & -2 \end{bmatrix}\), B = \(\begin{bmatrix} 3 & -1 & 2\\ 1 & 0 & 3\\ 2 & -1 & 0 \end{bmatrix}\) and M = \(\begin{bmatrix} 5 & 2\\ 1 & 4 \end{bmatrix}\), then
A + B = \(\begin{bmatrix} -1 & 2 & 3\\ 2 & -3 & 1\\ 3 & 1 & -2 \end{bmatrix}\) + \(\begin{bmatrix} 3 & -1 & 2\\ 1 & 0 & 3\\ 2 & -1 & 0 \end{bmatrix}\)
= \(\begin{bmatrix} -1 + 3 & 2 + (- 1) & 3 + 2\\ 2 + 1 & -3 + 0 & 1 + 3\\ 3 + 2 & 1 + (-1) & -2 + 0 \end{bmatrix}\)
= \(\begin{bmatrix} 2 & 1 & 5\\ 3 & -3 & 4\\ 5 & 0 & -2 \end{bmatrix}\)
A + M is not defined since the order of matrix M is not equal to the order of matrix A.
B + M is also not defined since the order of matrix M is not equal to the order of matrix B.
Note: Let A and B are m × n matrices and c, d are scalars. Then the following results are obvious.
I. c(A + B) = cA + cB,
For Example:
If A = \(\begin{bmatrix} 2 & 5\\ 3 & 1 \end{bmatrix}\) and B = \(\begin{bmatrix} 3 & 2\\ 0 & 7 \end{bmatrix}\) are m × n matrices and 5 is scalar. Then
\[5\left (\begin{bmatrix} 2 & 5\\ 3 & 1 \end{bmatrix} + \begin{bmatrix} 3 & 2\\ 0 & 7 \end{bmatrix} \right ) = 5\begin{bmatrix} 2 & 5\\ 3 & 1 \end{bmatrix} + 5 \begin{bmatrix} 3 & 2\\ 0 & 7 \end{bmatrix}\]
II. (c + d)A = cA + dA.
For Example:
If A = \(\begin{bmatrix} 2 & 5\\ 3 & 1 \end{bmatrix}\) be m × n matrix and 5, 3 are scalars. Then
\[\left (5 + 3\right )\begin{bmatrix} 2 & 5\\ 3 & 1 \end{bmatrix} = 5\begin{bmatrix} 2 & 5\\ 3 & 1 \end{bmatrix} + 3\begin{bmatrix} 2 & 5\\ 3 & 1 \end{bmatrix}\]
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