Question

How many 2 × 2 matrices A satisfy both A 3 = I2 and A2 = At , where I2 denotes the 2 × 2 identity matrix and At denotes the transpose of A ?​

Answer 1

Answer:

Let A=[

a

c

b

d

] where a,b,c,d∈R

tr(A)=a+d

Now, A

2

=[

a

c

b

d

][

a

c

b

d

]

⇒A

2

=[

a

2

+bc

ac+cd

ab+bd

bc+d

2

]

Given A

2

=I

⇒[

a

2

+bc

ac+cd

ab+bd

bc+d

2

]=[

1

0

0

1

]

⇒a

2

+bc=1=bc+d

2

and ab+bd=0=ac+cd

Since, A



=I and A



=−I

⇒a=−d

So,a=

1−bc

and d=−

1−bc

Hence, A=[

1−bc

c

b

−

1−bc

]

⇒∣A∣=

∣

∣

∣

∣

∣

∣

1−bc

c

b

−

1−bc

∣

∣

∣

∣

∣

∣

⇒∣A∣=−1

Statement 1 is true.

Here, tr(A)=

1−bc

−

1−bc

=0

Answer 2

Step-by-step explanation:

Let A=(

a

c

b

d

), abcd



=0

A

2

=(

a

c

b

d

)⋅(

a

c

b

d

)

A

2

=(

a

2

+bc

ac+cd

ab+bd

bc+d

2

)

=a

2

+ bc =1, bc +d

2

=1

ab+bd=ac+cd=0

c



=0 and b



=0=a+d=0

Trace A=a+d=0⇒a=−d

∣A∣=ad− bc =−a

2

− bc =−1

Hence, option 'B' is correct