The adjoint of the matrix {A} = ≤ft[ {array}{ccc} 1&0&2\\ 2&1&0\\ 0&3&1 {array} ] is

Correct Answer is : [16−2 −214 6−31]{bmatrix} 1 & 6 & -2 \ -2 & 1 & 4 \ 6 & -3 & 1 {bmatrix}[16−2 −214 6−31]

{bmatrix} -1 & 6 & 2 \\ -2 & 1 & -4 \\ 6 & 3 & 1 {bmatrix}

{bmatrix} 1 & 6 & -2 \ -2 & 1 & 4 \ 6 & -3 & 1 {bmatrix}

{bmatrix} 6 & 1 & 2 \\ 4 & -1 & 2 \\ 6 & 3 & -1 {bmatrix}

≤ft[ {array}{ccc} -6 & 2 & 1 \\ 4 & -2 & 1 \\ 3 & 1 & -6 {array} ]

Correct Answer is : [16−2 −214 6−31]{bmatrix} 1 & 6 & -2 \ -2 & 1 & 4 \ 6 & -3 & 1 {bmatrix}[16−2 −214 6−31]

Test Index

UPSC NDA 2 2017 Mathematics Solved Paper

Question : 97 of 120

Marks: +1, -0

\mathrm{A} = \left[ \begin{array}{ccc} 1&0&2\\ 2&1&0\\ 0&3&1 \end{array} \right]

$\begin{bmatrix} -1 & 6 & 2 \\ -2 & 1 & -4 \\ 6 & 3 & 1 \end{bmatrix}$
$\begin{bmatrix} 1 & 6 & -2 \ -2 & 1 & 4 \ 6 & -3 & 1 \end{bmatrix}$
$\begin{bmatrix} 6 & 1 & 2 \\ 4 & -1 & 2 \\ 6 & 3 & -1 \end{bmatrix}$
$\left[ \begin{array}{ccc} -6 & 2 & 1 \\ 4 & -2 & 1 \\ 3 & 1 & -6 \end{array} \right]$

Solution:

Concept:

The adjoint of a matrix is the transpose of its cofactor matrix.

Explanation:

First, find the minors of

A = \begin{bmatrix} 1 & 0 & 2 \\ 2 & 1 & 0 \\ 0 & 3 & 1 \end{bmatrix}

A minor

m_{ij}

is the determinant after removing the

i

-th row and

j

-th column.

m_{11} = (1)(1) - (0)(3) = 1

m_{12} = (2)(1) - (0)(0) = 2

m_{13} = (2)(3) - (1)(0) = 6

m_{21} = (0)(1) - (2)(3) = -6

m_{22} = (1)(1) - (2)(0) = 1

m_{23} = (1)(3) - (0)(0) = 3

m_{31} = (0)(0) - (2)(1) = -2

m_{32} = (1)(0) - (2)(2) = -4

m_{33} = (1)(1) - (0)(2) = 1

Now compute cofactors:

c_{ij} = (-1)^{i+j} m_{ij}

c_{11}=+1,\; c_{12}=-2,\; c_{13}=+6

c_{21}=+6,\; c_{22}=+1,\; c_{23}=-3

c_{31}=-2,\; c_{32}=+4,\; c_{33}=+1

So the cofactor matrix is

C = \begin{bmatrix} 1 & -2 & 6 \\ 6 & 1 & -3 \\ -2 & 4 & 1 \end{bmatrix}

The adjoint is the transpose:

\operatorname{adj}(A) = C^T = \begin{bmatrix} 1 & 6 & -2 \\ -2 & 1 & 4 \\ 6 & -3 & 1 \end{bmatrix}

Answer:

\begin{bmatrix} 1 & 6 & -2 \\ -2 & 1 & 4 \\ 6 & -3 & 1 \end{bmatrix}

which matches option B.

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