Find the inverse of the matrix {bmatrix} -2 & 4 \\ 3 & -5 {bmatrix}

Correct Answer is : [522321]{bmatrix} 5/2 & 2 \\ 3/2 & 1 {bmatrix}[252321]

{bmatrix} 5/2 & 2 \\ 1 & 3/2 {bmatrix}

{bmatrix} 2 & 5/2 \\ 3/2 & 1 {bmatrix}

{bmatrix} 5/2 & 2 \\ 3/2 & 1 {bmatrix}

{bmatrix} 2 & 5/2 \\ 1 & 3/2 {bmatrix}

Correct Answer is : [522321]{bmatrix} 5/2 & 2 \\ 3/2 & 1 {bmatrix}[252321]

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Question : 29 of 36

Marks: +1, -0

\begin{bmatrix} -2 & 4 \\ 3 & -5 \end{bmatrix}

Solution:

Concept:

The inverse of a

2 \times 2

matrix

\begin{bmatrix} a & b \\ c & d \end{bmatrix}

\frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}

, provided the determinant is non‑zero.

Explanation:

Let

A = \begin{bmatrix} -2 & 4 \\ 3 & -5 \end{bmatrix}

Compute the determinant:

ad - bc = (-2)(-5) - (4)(3) = 10 - 12 = -2

Form the adjugate:

\begin{bmatrix} d & -b \\ -c & a \end{bmatrix} = \begin{bmatrix} -5 & -4 \\ -3 & -2 \end{bmatrix}

Multiply by

\frac{1}{-2}

A^{-1} = -\frac{1}{2} \begin{bmatrix} -5 & -4 \\ -3 & -2 \end{bmatrix} = \begin{bmatrix} \frac{5}{2} & 2 \\ \frac{3}{2} & 1 \end{bmatrix}

Answer:

The inverse is

\begin{bmatrix} \frac{5}{2} & 2 \\ \frac{3}{2} & 1 \end{bmatrix}

, which corresponds to option C.

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