The inverse of the matrix ≤ft[{array}{ccc} 2 & 3 & 4 \\ 4 & 3 & 1 \\ 1 & 2 & 4 {array}] is

Correct Answer is : [10−4−9−154145−1−6]{bmatrix} 10 & -4 & -9 \\ -15 & 4 & 14 \\ 5 & -1 & -6 {bmatrix}10−155−44−1−914−6

{bmatrix} -2 & 4/5 & 9/5 \\ 3 & -4/5 & -14/5 \\ -1 & 1/5 & 6/5 {bmatrix}

{bmatrix} 10 & -4 & -9 \\ -15 & 4 & 14 \\ 5 & -1 & -6 {bmatrix}

{bmatrix} -10 & 4 & 9 \\ 15 & -4 & -14 \\ -5 & 1 & 6 {bmatrix}

Correct Answer is : [10−4−9−154145−1−6]{bmatrix} 10 & -4 & -9 \\ -15 & 4 & 14 \\ 5 & -1 & -6 {bmatrix}10−155−44−1−914−6

Test Index

GATE Civil Engineering (CE) 2019 Shift 2 Solved Paper

Question : 55 of 65

Marks: +1, -0

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

$\begin{bmatrix} -2 & \frac{4}{5} & \frac{9}{5} \\ 3 & -\frac{4}{5} & -\frac{14}{5} \\ -1 & \frac{1}{5} & \frac{6}{5} \end{bmatrix}$
$\begin{bmatrix} 10 & -4 & -9 \\ -15 & 4 & 14 \\ 5 & -1 & -6 \end{bmatrix}$
$\begin{bmatrix} -10 & 4 & 9 \\ 15 & -4 & -14 \\ -5 & 1 & 6 \end{bmatrix}$
$\begin{bmatrix} -10 & 4 & 9 \\ 15 & -4 & -14 \\ -5 & 1 & 6 \end{bmatrix}$

Solution:

A=\begin{array}{ccc} 2 & 3 & 4 \\ 4 & 3 & 1 \\ 1 & 2 & 4 \end{array}

Det. of matrix,

\lvert A \rvert = 2(12-2)-3(16-1)+4(8-3)

=20-45+20

=-5

A^{-1} = \frac{1}{\lvert A \rvert}(, A)

adj.

\text{adj} A = (\text{cof} A)'

A_{11} = 12-2 = 10

cof of

A_{12} = -(16-1) = -15

cof of

A_{13} = 8-3 = 5

cof of

A_{21} = -(12-8) = -4

cof of

A_{22} = (8-4) = 4

cof of

A_{23} = -(4-3) = -1

cof of

A_{31} = (3-12) = -9

cof of

A_{32} = -(2-16) = 14

cof of

A_{33} = (6-12) = -6

cof of

A = \begin{bmatrix} 10 & -15 & 5 \\ -4 & 4 & -1 \\ -9 & 14 & -6 \end{bmatrix}

cof

\text{adj} A = (\text{cof} A)'

= \begin{bmatrix} 10 & -4 & -9 \\ -15 & 4 & 14 \\ 5 & -1 & -6 \end{bmatrix}

A^{-1} = -\frac{1}{5} \begin{bmatrix} 10 & -4 & -9 \\ -15 & 4 & 14 \\ 5 & -1 & -6 \end{bmatrix}

= \begin{bmatrix} -2 & \frac{4}{5} & \frac{9}{5} \\ 3 & -\frac{4}{5} & -\frac{14}{5} \\ -1 & \frac{1}{5} & \frac{6}{5} \end{bmatrix}

\begin{bmatrix} 2 & -\frac{4}{5} & -\frac{9}{5} \\ -3 & \frac{4}{5} & \frac{14}{5} \\ 1 & -\frac{1}{5} & -\frac{6}{5} \end{bmatrix}

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