Q. 75.0( 2 Votes )

If and find

(i) X + Y

(ii) 2X – 3Y

(iii) A matrix Z such that X + Y + Z is a zero matrix.

Answer :

Addition or subtraction of matrices is possible only if the matrices are of same order.

That is,


If A and B are two matrices and if they are needed to be added, then if order of A is m × n, order of B must be m × n.


We have matrices X and Y, where




We know what order of matrix is,


If a matrix has M rows and N columns, the order of matrix is M × N.


(i). We need to find the X + Y.


Let us first determine order of X and Y.


Order of X:


Number of rows = 2


M = 2


Number of columns = 3


N = 3


Then, order of matrix X = M × N


Order of matrix X = 2 × 3


Order of Y:


Number of rows = 2


M = 2


Number of columns = 3


N = 3


Then, order of matrix Y = M × N


Order of matrix Y = 2 × 3


Since, order of matrix X = order of matrix Y


Matrices X and Y can be added.


So,





Thus, .


(ii). We need to find 2X – 3Y.


Let us calculate 2X.


We have,



Then, multiplying by 2 on both sides, we get





Also,



Multiplying by 3 on both sides, we get





Now subtract 3Y from 2X.





Thus, .


(iii). We need to find matrix Z, such that X + Y + Z is a zero matrix.


That is,


X + Y + Z = 0


Or,


Z = -X – Y


Or,


Z = -(X + Y)


We have already found X + Y in part (i).


So, from part (i):



Then,




Thus, .


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