Consider the matrix A= 3 1 -6 -2 , then Ι+A^40 is equal to

Correct answer: (D) Ι+2^40-1A. A^2= 3 1 -6 -2 3 1 -6 -2 = 3 1 -6 -2 =A A^3=A^4=A^5.........=A^40=A Ι+A^40=Ι+^40C_1A+^40C_2A^2.......+^40C_40A^40 =Ι+A^40C_1+^40C_2.......^40C_40 =Ι+A2^40-1

Q18 / 75 · NTA Abhyas (JEE Main) 2020 · Mathematics · Matrices

Consider the matrix $A = [\begin{array}{l} 3 & 1 \\ - 6 & - 2 \end{array}]$ , then ${(Ι + A)}^{40}$ is equal to

A
$Ι + 2^{38} A$
B
$Ι + 2^{39} A$
C
$Ι + (2^{40} + 1) A$
Correct answer
D
$Ι + (2^{40} - 1) A$

Solution

A^{2} = [\begin{array}{l} 3 & 1 \\ - 6 & - 2 \end{array}] [\begin{array}{l} 3 & 1 \\ - 6 & - 2 \end{array}]

= [\begin{array}{l} 3 & 1 \\ - 6 & - 2 \end{array}] = A

A^{3} = A^{4} = A^{5} . . . . . . . . . = A^{40} = A

{(Ι + A)}^{40} = Ι +^{40} C_{1} A +^{40} C_{2} A^{2} . . . . . . . +^{40} C_{40} A^{40}

= Ι + A [^{40} C_{1} +^{40} C_{2} . . . . . . .^{40} C_{40}]

= Ι + A (2^{40} - 1)

Hence, the correct answer is option: 'D'.

Consider the matrix A=31-6-2, then Ι+A40 is equal to

Solution

More Matrices Questions

Consider the matrix $A = [\begin{array}{l} 3 & 1 \\ - 6 & - 2 \end{array}]$ , then ${(Ι + A)}^{40}$ is equal to