Let A= ai j be a 3 × 3 matrix, where ai j= begincases(-1)j-1 text if i j, endcases then det(3 A d j(2 A-1)) is equal to .

Question

Let A= ai j be a 3 × 3 matrix, where ai j= begincases(-1)j-1 text if i < j, 2 text if i=j, (-1)i+j text if i>j, endcases then det(3 A d j(2 A-1)) is equal to . (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

A=[2 -1 1 -1 2 -1 1 -1 2] |A|=4 |3 a d j(2 A-1)|=|3.22 a d j(A-1)| =123| textadj(A-1)|=123|A-1|2 =(123/|A|2)=(123/16)=108

Q. Let $A = {a_{ij}}$ be a $3 \times 3$ matrix, where $a_{ij} = ⎩ ⎨ ⎧ (- 1)^{j - 1} 2 (- 1)^{i + j} if i < j, if i = j, if i > j,$ then det $(3 A d j (2 A^{- 1}))$ is equal to _______.

$A = ⎣ ⎡ 2 - 1 1 - 1 2 - 1 1 - 1 2 ⎦ ⎤$
$∣ A ∣ = 4$
$∣ ∣ 3 a d j (2 A^{- 1}) ∣ ∣ = ∣ ∣ 3. 2^{2} a d j (A^{- 1}) ∣ ∣$
$= 1 2^{3} ∣ ∣ adj (A^{- 1}) ∣ ∣ = 1 2^{3} ∣ ∣ A^{- 1} ∣ ∣^{2}$
$= \frac{1 2 ^{3}}{∣ A ∣ ^{2}} = \frac{1 2 ^{3}}{16} = 108$

Q. Let A={aij​} be a 3×3 matrix, where aij​=⎩⎨⎧​(−1)j−12(−1)i+j​ if i<j, if i=j, if i>j,​ then det(3Adj(2A−1)) is equal to _______.

A=⎣⎡​2−11​−12−1​1−12​⎦⎤​ ∣A∣=4 ∣∣​3adj(2A−1)∣∣​=∣∣​3.22adj(A−1)∣∣​ =123∣∣​adj(A−1)∣∣​=123∣∣​A−1∣∣​2 =∣A∣2123​=16123​=108

Q. Let $A = {a_{ij}}$ be a $3 \times 3$ matrix, where $a_{ij} = ⎩ ⎨ ⎧ (- 1)^{j - 1} 2 (- 1)^{i + j} if i < j, if i = j, if i > j,$ then det $(3 A d j (2 A^{- 1}))$ is equal to _______.

$A = ⎣ ⎡ 2 - 1 1 - 1 2 - 1 1 - 1 2 ⎦ ⎤$
$∣ A ∣ = 4$
$∣ ∣ 3 a d j (2 A^{- 1}) ∣ ∣ = ∣ ∣ 3. 2^{2} a d j (A^{- 1}) ∣ ∣$
$= 1 2^{3} ∣ ∣ adj (A^{- 1}) ∣ ∣ = 1 2^{3} ∣ ∣ A^{- 1} ∣ ∣^{2}$
$= \frac{1 2 ^{3}}{∣ A ∣ ^{2}} = \frac{1 2 ^{3}}{16} = 108$