Question
Download Solution PDFLet \(\rm M=\begin{bmatrix}4&-3\\\ 1&0\end{bmatrix}\)
Consider the following statements:
𝑃: 𝑀8 + 𝑀12 is diagonalizable.
𝑄: 𝑀7 + 𝑀9 is diagonalizable.
Which of the following statements is correct?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept -
(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.