Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa

Let A, other than \l or -\l, be a 2\times2 real matrix such that a^{2}=\l , \l being the unit matrix. Let Tr(A) be the sum of diagonal elements of A. 

Statement 1:Tr(A)=0

Statement 2:det(A)=-1

Option: 1

statement 1 is true; statement 2 is false.


Option: 2

statement 1 is true; statement 2 is true; statement 2 is not correct explanation for statement 1


Option: 3

statement 1 is true; statement 2 is true; statement 2 is correct explanation for statement 1


Option: 4

statement 1 is false; statement 2 is true; 


Answers (1)

best_answer

\begin{array}{l} {\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]} \\ {\left[\begin{array}{ll} a^{2}+b c & a b+b d \\ a c+c d & b c+d^{2} \end{array}\right]=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]} \end{array}

\begin{aligned} &b(a+d)=0, b=0 \text { or } a=-d\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ldots(1)\\ &c(a+d)=0, c=0 \text { or } a=-d\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ldots(2)\\ &a^{2}+b c=1, b c+d^{2}=1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ldots(3) \end{aligned}

'a' and 'd' are the diagonal element, a + d = 0

Statement 1 is true

Now, det(A) = ad - bc

\\\begin{aligned} &\text { Now, from (3) } a^{2}+b c=1 \text { and } d^{2}+b c=1\\ &\text { So, } a^{2}-d^{2}=0\\ &\text { Adding } a^{2}+d^{2}+2 b c=2\\ &=(a+d)^{2}-2 a d+2 b c=2 \end{aligned}\\\begin{aligned} &\text { or } 0-2(a d-b c)=2\\ &\text { So, } a d-b c=1 \Rightarrow \operatorname{det}(A)=-1 \end{aligned}
So, statement -2 is also true.
But statement -2 is not the correct explanation of statement-I 

Posted by

Riya

View full answer

Similar Questions

Latest Question