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Elementary Transformations

Suppose A i...

Question

Suppose $A$ is any $3 \times 3$ non-singular matrix and $(A - 3 I) (A - 5 I) = O$ , where $I = I_{3}$ and $O = O_{3}$ , If $α A + β A^{- 1} = 4 I$ , then $α + β$ is equal to :

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Solution

The correct option is A $8$
$\begin{matrix} (A - 3 I) (A - 5 I) = 0 \Rightarrow A^{2} - 8 A + 15 I = 0 \Rightarrow A - 8 I + 15 A^{- 1} = 0 \Rightarrow \frac{A}{2} + \frac{15}{2} A^{- 1} = 4 I α = \frac{1}{2} = β = \frac{15}{2} h e n c e, α + β = 8 \end{matrix}$
hence, option $A$ is correct.

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Similar questions

Q. Suppose

A

is any

3 \times 3

non-singular matrix and

(A - 3 I) (A - 5 I) = O

, where

I = I_{3}

and

O = O_{3}

. If

α A + β A^{- 1} = 4 I

, then

α + β

is equal to :

Q. Suppose

A

is any

3 \times 3

non-singular matrix and

(A - 3 I) (A - 5 I) = O

, where

I = I_{3}

and

O = O_{3}

. If

α A + β A^{- 1} = 4 I

, then

α + β

is equal to :

Q. If

A

is a non-singular matrix such that

(A - 2 I) (A - 4 I) = O

, where

I

is an identity matrix of the same order as that of

A

. Then

\frac{1}{6} A + \frac{4}{3} A^{- 1}

is equal to:

Q. Let

P

be a non-singular matrix and

I + P + P^{2} + \dots + P^{n} = O

. Then

P^{- 1}

is equal to (where

n \in Z^{+}

)

Q. If matrix

A

is non-singular and satisfies

A^{2} - A + I = O,

then the inverse of

A

is equal to

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Suppose A is any 3×3 non-singular matrix and (A−3I)(A−5I)=O, where I=I3 and O=O3, If αA+βA−1=4I, then α+β is equal to :

The correct option is A 8(A−3I)(A−5I)=0⇒A2−8A+15I=0⇒A−8I+15A−1=0⇒A2+152A−1=4Iα=12=β=152hence,α+β=8hence, option A is correct.

Suppose $A$ is any $3 \times 3$ non-singular matrix and $(A - 3 I) (A - 5 I) = O$ , where $I = I_{3}$ and $O = O_{3}$ , If $α A + β A^{- 1} = 4 I$ , then $α + β$ is equal to :

The correct option is A $8$
$\begin{matrix} (A - 3 I) (A - 5 I) = 0 \Rightarrow A^{2} - 8 A + 15 I = 0 \Rightarrow A - 8 I + 15 A^{- 1} = 0 \Rightarrow \frac{A}{2} + \frac{15}{2} A^{- 1} = 4 I α = \frac{1}{2} = β = \frac{15}{2} h e n c e, α + β = 8 \end{matrix}$
hence, option $A$ is correct.