Chapter 3 Matrices Ex 3.2 Solutions
Question - 21 : - Assume┬аX,┬аY,┬аZ,┬аW┬аand┬аP┬аarematrices of order
, and┬а
respectively. The restrictionon┬аn,┬аk┬аand┬аp┬аso that┬а
will be defined are:
Answer - 21 : -
A.┬аk┬а= 3,┬аp┬а=┬аn┬а ┬а ┬а ┬а ┬а ┬а ┬а ┬а ┬а ┬аB.┬аk┬аis arbitrary,┬аp┬а= 2
C.┬аp┬аis arbitrary,┬аk┬а= 3┬а ┬а ┬а ┬а┬аD.┬аk┬а= 2,┬аp┬а= 3
Solution
Matrices┬аP┬аand┬аY┬аareof the orders┬аp┬а├Ч┬аk┬аand 3 ├Ч┬аk┬аrespectively.
Therefore, matrix┬аPY┬аwillbe defined if┬аk┬а= 3. Consequently,┬аPY┬аwill beof the order┬аp┬а├Ч┬аk.
Matrices┬аW┬аand┬аY┬аareof the orders┬аn┬а├Ч 3 and 3 ├Ч┬аk┬аrespectively.
Since the number ofcolumns in┬аW┬аis equal to the number of rows in┬аY,matrix┬аWY┬аis well-defined and is of the order┬аn┬а├Ч┬аk.
Matrices┬аPY┬аand┬аWY┬аcanbe added only when their orders are the same.
However,┬аPY┬аisof the order┬аp┬а├Ч┬аk┬аand┬аWY┬аisof the order┬аn┬а├Ч┬аk. Therefore, we must have┬аp┬а=┬аn.
Thus,┬аk┬а=3 and┬аp┬а=┬аn┬аare the restrictions on┬аn,┬аk,and┬аp┬аso that┬аwill be defined.
Question - 22 : - Assume┬аX,┬аY,┬аZ,┬аW┬аand┬аP┬аarematrices of order┬а
, and┬а
respectively.┬а
Answer - 22 : - If┬аn┬а=┬аp, then the order of the matrix┬а
is┬а
┬аA┬аp┬а├Ч┬а2┬а ┬а ┬а ┬а ┬а┬аB┬а2┬а├Ч┬аn┬а ┬а ┬а ┬а ┬а ┬аC┬аn┬а├Ч┬а3┬а ┬а ┬а ┬а ┬а ┬а┬аD┬аp┬а├Ч┬аn
Solution
The correct answer is B.
Matrix┬аX┬аisof the order 2┬а├Ч┬аn.
Therefore, matrix 7X┬аisalso of the same order.
Matrix┬аZ┬аisof the order 2┬а├Ч┬аp, i.e., 2┬а├Ч┬аn┬а[Since┬аn┬а=┬аp]
Therefore, matrix 5Z┬аisalso of the same order.
Now, both the matrices 7X┬аand5Z┬аare of the order 2┬а├Ч┬аn.
Thus, matrix 7X┬атИТ5Z┬аis well-defined and is of the order 2┬а├Ч┬аn.