Question

Assume X, Y, Z, W and P are matrices of order \[2\times n,3\times k,2\times p,n\times 3,p\times k\] respectively. The restriction on n, k and p, so that PY + WY will be defined are:
(a) k = 3, p = n
(b) k is arbitrary, p = 2
(c) p is arbitrary, k = 3
(d) k = 2, p = 3

Answer 1

Hint: To solve this question, we will, first of all, observe when a matrix multiplication is defined. Two matrix product AB is defined when the number of columns of A is equal to the number of rows of B. Also, if a matrix is of the order \[m\times n\] then m is the number of rows and n is the number of columns. Using this, we will see when PY + WY is defined.

Complete step by step answer:
We are given that X is a matrix of order \[2\times n.\]
Y is the matrix of order \[3\times k.\]
Z is the matrix of order \[2\times p.\]
W is the matrix of order \[n\times 3.\]
P is the matrix of order \[p\times k.\]
We want PY + WY to be well defined.
Matrix multiplication between two matrices A and B is defined if the number of columns of matrix A is equal to the number of rows of matrix B, then AB would be defined. Consider PY first. For PY to be defined by the above fact, the number of columns of matrix P should be equal to the number of rows of matrix Y. If a matrix M is of the order \[t\times s\] then t is the number of rows and s is the number of columns.
PY is defined in the column of P = rows of Y
\[\Rightarrow k=3\]
So, the value of k should be 3…..(i).
Again for WY to be defined we should have by the above logic that the number of columns of W = number of rows of Y.
\[\Rightarrow 3=3\]
Hence, which is correct.
So, PY + WY is well defined, as the sum of well-defined matrices is always well – defined.
Hence, the option (c) is the right answer.

Note:
The biggest possibility of mistake can be considering the P matrix too, to get the value of ‘p’. This is not required as PY + WY is well defined when PY and WY are so. And for PY to be well defined, we only need columns of P and not rows of P. So, there is no need of ‘p’.