Let M4-I- Where i denotes the identity matrix and M- I and M2- I and M3- I- Then- for any natural number k- K-1 equals-

As we know that nbsp; MM^ 1 =I Let nbsp;M^ 1 = nbsp;M^4k+n [as per options given] ⇒ M.M^4k+n =I ⇒ M.M^4k.M^n =I ⇒ M.(M^4)^k.M^n =I ⇒ M^n+1.I nbsp; =I ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Other

Engineering Mathematics

Inverse of matrix

Let M4=I, Whe...

Question

Let $M^{4} = I$ , (Where i denotes the identity matrix) and $M \neq I$ and $M^{2} \neq I$ and $M^{3} \neq I .$ Then, for any natural number k, $K^{- 1}$ equals:

M^{4 k + 1}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

M^{4 k + 2}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

M^{4 k + 3}

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

M^{4 k}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is C $M^{4 k + 3}$

As we know that
MM $^{- 1}$ =I
Let M $^{- 1}$ = M $^{4 k + n}$ [as per options given]
$\Rightarrow M . M^{4 k + n} = I$
$\Rightarrow M . M^{4 k} . M^{n} = I$
$\Rightarrow M . (M^{4})^{k} . M^{n} = I$
$\Rightarrow M^{n + 1} . I = I$ [ $∵$ M^{4} =1-given )]

Now, if n = 3, then $M^{n + I} = M^{4} = 1$
Hence, $M^{4} .1 = 1$
$\Rightarrow I . I = I$ $∵ n = 3$
$\Rightarrow$ identity is satisfied
Hence, (c)

Method II:
$∵ M^{4} = I$
i.e., $M . M^{3} = I$ (given)
But we know that
$M M^{- 1} = I$

Hence on comparison
$M^{- 1} = M^{3}$
so for K = 1, only option (c)
satisfies above condition.

Suggest Corrections

Similar questions

Q. If

m_{1}, m_{2}, m_{3}

and

m_{4}

are respectively the magnitudes of the vectors

a_{1} = 2 i - j + k, a_{2} = 3 i - 4 j - 4 k,

a_{3} = i + j - k

and

a_{4} = - i + 3 j + k,

then the correct order of

m_{1}, m_{2}, m_{3}

and

m_{4}

Q. Let A denote the matrix

[\begin{matrix} 0 & i i & 0 \end{matrix}]

, where

i^{2} = - 1

and lit I denote the identity matrix

[\begin{matrix} 0 & i i & 0 \end{matrix}]

.Then

I + A + A^{2} + . . . . . A^{2010}

Q. If

m_{1}, m_{2}, m_{3}, m_{4}

, are respectively the magnitudes of the vectors

_{1} = ¯ ¯¯¯ ¯ 2 i - ¯ j + ¯ ¯ ¯ k,_{2} = ¯ ¯¯¯ ¯ 3 i - ¯ ¯¯¯ ¯ 4 j - ¯ ¯¯¯¯ ¯ 4 k,_{3} = - ¯ i + ¯ j - ¯ ¯ ¯ k,_{4} = - ¯ i + 3 ¯ j + ¯ ¯ ¯ k

, then the correct order of

m_{1}, m_{2}, m_{3}, m_{4}

Q. Let

M

be a

3 \times 3

invertible matrix with real entries and let

I

denote the

3 \times 3

identity matrix. If

M^{- 1} = a d j (a d j M),

then which of the following options is (are) always TRUE?

Q. Let

I

denote the

3 \times 3

identity matrix and

P

be a matrix obtained by rearranging the columns of

I

. Then

Join BYJU'S Learning Program

Let M4=I, (Where i denotes the identity matrix) and M≠I and M2≠I and M3≠I. Then, for any natural number k, K−1 equals:

Let $M^{4} = I$ , (Where i denotes the identity matrix) and $M \neq I$ and $M^{2} \neq I$ and $M^{3} \neq I .$ Then, for any natural number k, $K^{- 1}$ equals: