Question

Let $A=\left[a_{ij}\right]$ be a square matrix of order 2 where $a_{ij}ϵ\{0,1,2,3,4,6\}.$ The number of matrices A with distinct element such that $A{A}^{-1}=I$, where I is the unit matrix of order 2, is $(a^3+1).$ Find the value of a.___

Answer 1

Since $A^{-1}$ exists.

$\therefore$ A must be non-singular

Case I: When one of the elements is zero

Total no. of matrices $={}^5{C}_3(4!)=240$

Case II: When all non-zero distinct digits are used.
Let $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

|A| = ad - bc

Counting those ways when |A| = 0, i.e. ad = bc.

$\begin{array}{ccccc}\text{a}& \text{b}& \text{c}& \text{d}& \text{Total ways}\\ \text{1}& \text{2}& \text{3}& \text{6}& \text{4 ways}\\ \text{2}& \text{1}& \text{6}& \text{3}& \text{4 ways}\\ \text{2}& \text{3}& \text{4}& \text{6}& \text{4 ways}\\ \text{3}& \text{2}& \text{6}& \text{4}& \text{4 ways}\\ & & & \text{Total}& \text{= 16 ways}\end{array}$

Number of matrices A when |A| = 0 is 16.

${\therefore }^5{C}_4(4!)-16=104$

Total number of matrices when

$A^{-1}\text{exists is}240+104=344.$

$a^3+1=344$

$a^3=343$

$a=7$