Question

The inverse of a matrix A is given by \(\rm \begin{bmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{bmatrix}\) What is A equal to?
1. \(\rm \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\)
2. \(\rm \begin{bmatrix} 1 & -2 \\ -3 & 4 \end{bmatrix}\)
3. \(\rm \begin{bmatrix} 1 & 2 \\ 3 & -4 \end{bmatrix}\)
4. \(\rm \begin{bmatrix} -1 & 2 \\ 3 & 4 \end{bmatrix}\)

Answer 1

Correct Answer - Option 1 : \(\rm \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\)

Concept:

The determinant of the inverse of an invertible matrix is the

inverse of the determinant:

det(A^-1) = 1 / det(A)

Calculation:

A^-1 = \(\begin{bmatrix} -\frac{1}{2} & -1 \\ \frac{-3}{2} & -2 \end{bmatrix}\)

Det(A^-1) = (-2) × \(\rm \frac{-1}{2}\)  -(\(\rm \frac{-3}{2}\) × 1) = \(\rm \frac{-1}{2}\)

⇒ Det(A-1) = \(\rm \frac{-1}{2}\)

⇒ A = -2        [∵ det(A^-1) = 1 / det(A)]

From the given option,

 \(\left| {\begin{array}{*{20}{c}} {1 }&{2 }\\ { 3 }&{4 } \end{array}} \right| \) = 4 - 6 = -2 

∴ Matrix A = \(\rm \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\)