If a matrix has 24 elements, what are possible order it can have? What, if it has 13 elements ?

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3 and 4 .Determinants and Matrices

easy

If a matrix has $24$ elements, what are the possible order it can have? What, if it has $13$ elements ?

Option A

Option B

Option C

Option D

Solution

We know that if a matrix is of the order $m \times n,$ it has $mn$ elements.

Thus, to find all the possible orders of a matrix having $24$ elements, we have to find all the ordered pairs of natural numbers whose product is $24$.

The ordered pairs are: $(1,24)$, $( 24,1)$, $(2,12)$, $(12,2)$, $(3,8)$, $(8,3)$, $(4,6),$ and $(6,4)$

Hence, the possible orders of a matrix having $24$ elements are:

$1 \times 24$, $24 \times 1$, $2 \times 12$, $12 \times 2$,

$3 \times 8$, $8 \times 3$ ,$4 \times 6$ and $6 \times 4$

$(1,13)$ and $(13,1)$ are the ordered pairs of natural numbers whose product is $13$.

Hence the possible orders of matrix having $13$ elements are $1 \times 13$ and $13 \times 1 .$

Standard 12

Mathematics

Show that

$\left[ {\begin{array}{{20}{l}}
1&2&3 \\
0&1&0 \\
1&1&0
\end{array}} \right]\left[ {\begin{array}{{20}{c}}
{ – 1}&1&0 \\
0&{ – 1}&1 \\
2&3&4
\end{array}} \right]$ $ \ne \left[ {\begin{array}{{20}{c}}
{ – 1}&1&0 \\
0&{ – 1}&1 \\
2&3&4
\end{array}} \right]\left[ {\begin{array}{{20}{l}}
1&2&3 \\
0&1&0 \\
1&1&0
\end{array}} \right]$

medium

If $A$ is square matrix such that $A^{2}=A$, then $(1+A)^{3}-7 A$ is equal to

medium

If the matrix $\left[ {\begin{array}{*{20}{c}}0&1&{ – 2}\\{ – 1}&0&3\\\lambda &{ – 3}&0\end{array}} \right]$ is singular, then $\lambda $=

easy

The number of elements in the set $\left\{A=\left(\begin{array}{ll}a & b \\ 0 & d\end{array}\right): a, b, d \in\{-1,0,1\}\right.$ and $\left.(I-A)^{3}=I-A^{3}\right\}$ where $I$ is $2 \times 2$ identity matrix, is :

hard

(JEE MAIN-2021)

If matrix $A = \left[ {\begin{array}{*{20}{c}}0&{ – 1}\\1&0\end{array}} \right]$, then ${A^{16}} = $

easy

If a matrix has $24$ elements, what are the possible order it can have? What, if it has $13$ elements ?

Solution

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