Question

If \(|A|=3\)  and \(A^{-1}=\begin{bmatrix}        3 &-1 \\[0.3em]        \frac{-5}{3} & \frac{2}{3}    \\[0.3em]     \end{bmatrix}\) then adj \(A\) is 

• A. \(\begin{bmatrix}    9 & 3  \\[0.3em]    5  &2  \end{bmatrix}\)
• B. \(\begin{bmatrix}    -9 & 3  \\[0.3em]    -5  &2  \end{bmatrix}\)
• C. \(\begin{bmatrix}    9 & -3  \\[0.3em]    -5  &2  \end{bmatrix}\)
• D. \(\begin{bmatrix}    -9 & 3  \\[0.3em]    5  &-2  \end{bmatrix}\)

Answer 1

The Correct Option is C

To find the adjugate (adj) of matrix \(A\), we use the relationship between the inverse of the matrix, its determinant, and its adjugate. Given:

• \(|A|=3\), the determinant of \(A\).
• \(A^{-1}=\begin{bmatrix}3 & -1 \\ -\frac{5}{3} & \frac{2}{3} \end{bmatrix}\).

The inverse of a matrix \(A\) is expressed as:

\(A^{-1}=\frac{1}{|A|}\cdot \text{adj}(A)\)

Rearranging gives us:

\(\text{adj}(A)=|A| \cdot A^{-1}\)

Substitute the given values:

\(\text{adj}(A) = 3 \cdot \begin{bmatrix}3 & -1 \\ -\frac{5}{3} & \frac{2}{3} \end{bmatrix}\)

Perform the scalar multiplication:

\(\text{adj}(A) = \begin{bmatrix}3 \times 3 & 3 \times (-1) \\ 3 \times (-\frac{5}{3}) & 3 \times \frac{2}{3} \end{bmatrix}\)

Simplify each element:

\(\text{adj}(A) = \begin{bmatrix}9 & -3 \\ -5 & 2 \end{bmatrix}\)

The adjugate of matrix \(A\) is \(\begin{bmatrix}9 & -3 \\ -5 & 2 \end{bmatrix}\).