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    ​The inverse of the matrix A=(11313−3−2−4−4)A=\left(\begin{array}{ccc}1 & 1 & 3 \\1 & 3 & -3 \\-2 & -4 & -4\end{array}\ri
    Question

    The inverse of the matrix A=(113133244)A=\left(\begin{array}{ccc}1 & 1 & 3 \\1 & 3 & -3 \\-2 & -4 & -4\end{array}\right)​ is:

    A.

    (311.51.250.250.750.250.250.25)\left(\begin{array}{ccc}3 & 1 & 1.5 \\-1.25 & -0.25 & -0.75 \\-0.25 & -0.25 & -0.25\end{array}\right)​​

    B.

    (311.51.500.250.750.250.250.25)\left(\begin{array}{ccc}3 & 1 & 1.5 \\-1.50 & -0.25 & -0.75 \\-0.25 & -0.25 & -0.25\end{array}\right)​​

    C.

    (311.51.250.750.250.250.250.25)\left(\begin{array}{ccc}3 & 1 & 1.5 \\-1.25 & -0.75 & -0.25 \\-0.25 & -0.25 & -0.25\end{array}\right)​​

    D.

    (311.51.250.250.750.250.750.25)\left(\begin{array}{ccc}3 & 1 & 1.5 \\-1.25 & -0.25 & -0.75 \\-0.25 & -0.75 & -0.25\end{array}\right)​​

    Correct option is A

    First, ensure the matrix is invertible by checking det(A)0.det(A)=1341212314=1(3(4)1(4))1((4)(3)(2))+3((1(4)3(2)))=1(12+4)1(46)+3(4+6)=8+10+6=8(Non-zero, so invertible)Find the Adjugate (Adjoint) of ACompute the cofactor matrix and transpose it.Cofactor matrix=(2410212628126)Transpose (Adjugate):adj(A)=(2412810612226)Compute the InverseA1=1det(A)adj(A)=18(2412810612226)=(31.511.250.751.50.250.250.75)\begin{aligned}&\text{First, ensure the matrix is invertible by checking } \det(A) \ne 0. \\&\det(A) = \begin{vmatrix} 1 & 3 & -4 \\ \frac{1}{2} & -1 & 2 \\ 3 & 1 & 4 \end{vmatrix} \\&= 1 \cdot (3 \cdot (-4) - 1 \cdot (-4)) - 1 \cdot ((-4) - (-3) \cdot (-2)) + 3 \cdot ((1 \cdot (-4) - 3 \cdot (-2))) \\&= 1 \cdot (-12 + 4) - 1 \cdot (-4 - 6) + 3 \cdot (-4 + 6) \\&= -8 + 10 + 6 = 8 \quad \text{(Non-zero, so invertible)} \\\\&\textbf{Find the Adjugate (Adjoint) of } A \\&\text{Compute the cofactor matrix and transpose it.} \\&\text{Cofactor matrix} = \begin{pmatrix}-24 & 10 & 2 \\12 & 6 & 2 \\-8 & -12 & 6\end{pmatrix} \\&\text{Transpose (Adjugate):} \\&\text{adj}(A) = \begin{pmatrix}-24 & 12 & -8 \\10 & 6 & -12 \\2 & 2 & 6\end{pmatrix} \\\\&\textbf{Compute the Inverse} \\&A^{-1} = \frac{1}{\det(A)} \cdot \text{adj}(A) = \frac{1}{8} \cdot\begin{pmatrix}-24 & 12 & -8 \\10 & 6 & -12 \\2 & 2 & 6\end{pmatrix} \\&= \begin{pmatrix}-3 & 1.5 & -1 \\1.25 & 0.75 & -1.5 \\0.25 & 0.25 & 0.75\end{pmatrix}\end{aligned}​​

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