Let M be a 3×3 invertible matrix with real entries and let I denote the 3×3 identity matrix. M^(−1) =adj adj M , then which of the following…

Question

Let M be a 3×3 invertible matrix with real entries and let I denote the 3×3 identity matrix. M^(−1) =adj adj M , then which of the following statements is/are ALWAYS TRUE?

MARKS Content Team · Accepted Answer

Correct answer: (B) detM=1; (C) M^2 =I; (D) adj M^2 =I. Given M^(−1)=adjadjM ∴ adj M⋅M^(−1)=adj M . adj adjM ⇒adj M.M^(−1)=AdjMI ⇒adj M=M^2M .........1 ⇒adj M=M^2M=M^6M ⇒M^2=M^7⇒M≠0, M=1 .............2 By equations 1& 2, we get adj M=M ...........3 ⇒M.adjM=M^2 ⇒MI=M^2⇒M^2=I Now, by equation 3, adj M=M ⇒adj M^2=M^2=I

Let $M$ be a $3 \times 3$ invertible matrix with real entries and let $I$ denote the $3 \times 3$ identity matrix. $M^{- 1} = a d j (a d j M)$ , then which of the following statements is/are ALWAYS TRUE?

Solution

More Matrices Questions

Let M be a 3×3 invertible matrix with real entries and let I denote the 3×3 identity matrix. M −1 =adj adj M , then which of the following statements is/are ALWAYS TRUE?

Solution

More Matrices Questions

Let $M$ be a $3 \times 3$ invertible matrix with real entries and let $I$ denote the $3 \times 3$ identity matrix. $M^{- 1} = a d j (a d j M)$ , then which of the following statements is/are ALWAYS TRUE?