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Two matrices A and B are said to be conformable for subtraction if they have the same order (i.e. same number of rows and columns) and their difference A - B is defined to be the addition of A and (-B).
i.e., A β B = A + (-B)
For example:
[a11a12a13a21a22a23a31a32a33] - [b11b12b13b21b22b23b31b32b33]
= [a11a12a13a21a22a23a31a32a33] + [βb11βb12βb13βb21βb22βb23βb31βb32βb33]
= [a11βb11a12βb12a13βb13a21βb21a22βb22a23βb23a31βb31a32βb32a33βb33]
Again, if A = (aij)m, n and B = (bij)m, n then their difference 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 = [a11a12a13a21a22a23a31a32a33] and B = [b11b12b13b21b22b23b31b32b33], then
A - B = [a11βb11a12βb12a13βb13a21βb21a22βb22a23βb23a31βb31a32βb32a33βb33] = C
Note: If A and B be matrices of different orders, then A - B is not defined.
Example on Subtraction of Matrices:
1. If A = [1231] and B = [2413], then
A - B = [1231] - [2413]
= [1β22β43β11β3]
= [β1β22β2]
2. If A = [0122β311β20], B = [β102321β2β10] and M = [4213], then
A - B = [0122β311β20] - [β102321β2β10]
= [0β11β02β22β3β3β21β11β(β2)β2β(β1)0β0]
= [β110β1β503β10]
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 Γ n matrices and c, d are scalars. Then the following results are obvious.
I. c(A - B) = cA - cB,
For Example:
If A = [1324] and B = [2130] are m Γ n matrices and 4 is scalar. Then
4([1324]β[2130])=4[1324]β4[2130]
II. (c - d)A = cA - dA.
For Example:
If A = [20β15] be m Γ n matrix and 4, 2 are scalars. Then
(4β2)[20β15]=4[20β15]β2[20β15]
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