Subtraction of Matrices

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 – B = A + (-B)

For example:

[a11a12a13a21a22a23a31a32a33] - [b11b12b13b21b22b23b31b32b33]

= [a11a12a13a21a22a23a31a32a33] + [βˆ’b11βˆ’b12βˆ’b13βˆ’b21βˆ’b22βˆ’b23βˆ’b31βˆ’b32βˆ’b33]

= [a11βˆ’b11a12βˆ’b12a13βˆ’b13a21βˆ’b21a22βˆ’b22a23βˆ’b23a31βˆ’b31a32βˆ’b32a33βˆ’b33]

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 = [a11βˆ’b11a12βˆ’b12a13βˆ’b13a21βˆ’b21a22βˆ’b22a23βˆ’b23a31βˆ’b31a32βˆ’b32a33βˆ’b33] = 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]

         = [1βˆ’22βˆ’43βˆ’11βˆ’3]

         = [βˆ’1βˆ’22βˆ’2]


2. If A = [0122βˆ’311βˆ’20], B = [βˆ’102321βˆ’2βˆ’10] and M = [4213], then 

A - B = [0122βˆ’311βˆ’20] - [βˆ’102321βˆ’2βˆ’10] 

        = [0βˆ’11βˆ’02βˆ’22βˆ’3βˆ’3βˆ’21βˆ’11βˆ’(βˆ’2)βˆ’2βˆ’(βˆ’1)0βˆ’0]

       = [βˆ’110βˆ’1βˆ’503βˆ’10]

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]βˆ’[2130])=4[1324]βˆ’4[2130]


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

For Example:

If A = [20βˆ’15] be m Γ— n matrix and 4, 2 are scalars. Then 

(4βˆ’2)[20βˆ’15]=4[20βˆ’15]βˆ’2[20βˆ’15]





10th Grade Math

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