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 $5 \times 1$.