Let \(\rm M=\begin{bmatrix}4&-3\\\ 1&0\end{bmatrix}\)

Consider the following statements:

𝑃: 𝑀⁸ + 𝑀¹² is diagonalizable.

𝑄: 𝑀⁷ + 𝑀⁹ is diagonalizable.

Which of the following statements is correct?

Concept -

(1) If a is the eigen value of M then the eigen value of Mⁿ is aⁿ.

(2) If all the eigen value of the matrix is different then the matrix is diagonalizable.

Explanation -

Given - \(\rm M=\begin{bmatrix}4&-3\\\ 1&0\end{bmatrix}\)

Now characteristic equation for the given matrix is -

⇒ | M - λ I | = 0

⇒ \(\rm det\begin{bmatrix}4-λ&-3\\\ 1&-λ\end{bmatrix}=0\)

⇒ -λ (4 - λ ) + 3 = 0 ⇒ λ² - 4λ + 3 = 0

Now solve this equation we get the eigen values of the matrix -

 ⇒ λ2 - 3λ - λ  + 3 = 0  ⇒ (λ -3)(λ -1) = 0  ⇒ λ = 1, 3 

So the eigen value of M are 1 and 3.

Now solve the given statements -

(P) 𝑀8 + 𝑀12 is diagonalizable.

Now the eigen value of 𝑀8 + 𝑀12  are  \((1)^8 + (1)^{12} = 2 \ \ and \ \ (3)^8 +(3)^{12} =82(3)^8\)

Hence both the eigen value of 𝑀8 + 𝑀12  are different So 𝑀8 + 𝑀12  is diagonalizable.

(𝑄) 𝑀7 + 𝑀9 is diagonalizable.

Now the eigen value of 𝑀7 + 𝑀9  are  \((1)^7 + (1)^9 = 2 \ \ and \ \ (3)^7 +(3)^9 =10(3)^7\)

Hence both the eigen value of 𝑀7 + 𝑀9 are different So 𝑀7 + 𝑀9 is diagonalizable.

Hence the option (4) is true.