Addition of Matrices | Example on Sum of Two Matrices

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We proceed to develop the algebra of
matrices.

Two matrices A and B are said to be
conformable for addition if they have the same order (same number of rows and
columns).

If A 
= (aij)m, n and B = (bij)m, n
then their sum A + B is the matrix C = (cij)m,n where cij
= aij + bij, 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})

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 + 7end{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 C is not equal to the order of matrix A.

B + M is also not defined since the order of matrix C is not equal to the order of matrix B.

Note: Let A be an m × n matrix and c, d are scalars. Then the following results are obvious. 

I. c(A + B) = cA + cB,

II. (c + d)A = cA + dA.

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10th Grade Math

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