Question

Consider the following statements.
I. If a matrix has $24$ elements, then all the possible orders it can have are $24\times 1,1\times 24,2\times 4,4\times 2,2\times 12,12$
$\times 2,3\times 8,8\times 3,4\times 6$ and $6\times 4$
II. For a matrix having $13$ elements, its all possible orders are $1\times 13$ and $13\times 1$.
III. For a matrix having $18$ elements, its all possible orders are $18\times 1,1\times 18,2\times 9,9\times 2,3\times 6,6\times 3$.
IV. For a matrix having $5$ elements, its all possible orders are $1\times 5$ and $5\times 1$. Choose the correct option

• A. Only I is false
• B. Only II is a false
• C. Only III is false
• D. All are true

Answer 1

Answer: (A)

If a matrix is of order $m\times n$, then it has $mn$ elements.
I. Thus, to find the all possible orders of a matrix with $24$ elements, we will find all ordered pairs of natural numbers, whose product is $24$.
Thus all possible order pairs are
$(1,24),(24,1),(2,12),(12,2),(3,8),(8,3),(4,6),(6,4)$
$\therefore$ All possible orders are
$1\times 24,24\times 1,2\times 12,12\times 2,3\times 8,8\times 3,4\times 6,6\times 4$
II. Similarly, if a matrix has 13 elements, then its all possible orders are $1\times 13$ and $13\times 1$.
III. If a matrix has 18 elements, the all its possible orders are
$18\times 1,1\times 18,2\times 9,9\times 2,3\times 6,6\times 3$
IV. A matrix have 5 elements, then its possible orders are $1\times 5$ and $5\times 1$.