Let A=[ -1 2 -3 -2 0 3 3 -3 1 ] be a matrix, then |A|adj (A- 1) is equal to

Question

Let A=[ -1 2 -3 -2 0 3 3 -3 1 ] be a matrix, then |A|adj (A- 1) is equal to (A) O3 × 3 (B) [ -1 2 -3 -2 0 3 3 -3 1 ] (C) I3 (D) [ -3 -3 &nbsp&nbsp&nbsp1 &nbsp&nbsp&nbsp3 &nbsp&nbsp&nbsp0 -2 -1 &nbsp&nbsp&nbsp2 -3 ]

Tardigrade Admin · Accepted Answer

We know that, A- 1adj (A- 1)= |A- 1|I3 ⇒ A. A- 1adj (A- 1)= |A- 1|(A I)3 ⇒ adj (A- 1)= |A- 1|A ⇒ |A|adj (A- 1)= A

Q. Let $A = ⎣ ⎡ - 1 - 2 3 20 - 3 - 3 31 ⎦ ⎤$ be a matrix, then $∣ A ∣ a d j (A^{- 1})$ is equal to

$O_{3 \times 3}$

$⎣ ⎡ - 1 - 2 3 20 - 3 - 3 31 ⎦ ⎤$

$I_{3}$

$⎣ ⎡ - 3 - 1 - 3 nb s p nb s p nb s p nb s p nb s p 3 nb s p nb s p nb s p 2 nb s p 1 nb s p - 3 nb s p nb s p 0 - 2 ⎦ ⎤$

We know that,
$A^{- 1} a d j (A^{- 1}) = ∣ ∣ A^{- 1} ∣ ∣ I_{3}$
$\Rightarrow A . A^{- 1} a d j (A^{- 1}) = ∣ ∣ A^{- 1} ∣ ∣ (A I)_{3}$
$\Rightarrow a d j (A^{- 1}) = ∣ ∣ A^{- 1} ∣ ∣ A$
$\Rightarrow ∣ A ∣ a d j (A^{- 1}) = A$

Q. Let A=⎣⎡​−1−23​20−3​−331​⎦⎤​ be a matrix, then ∣A∣adj(A−1) is equal to

O3×3​

⎣⎡​−1−23​20−3​−331​⎦⎤​

I3​

⎣⎡​−3−1​−3nbsp​nbspnbsp​nbspnbsp3nbsp​nbspnbsp2​nbsp1nbsp−3​nbsp​nbsp0​−2​⎦⎤​