Question

If the matrix \(A=\begin{bmatrix} 1 & 2 & 3 \\ 1 & 1 & 5 \\ 2 & 4 & 7 \end{bmatrix}\) and its inverse is \(A^{-1}=\begin{bmatrix} a & c & -7 \\ b & -1 & c \\ -2 & 0 & 1 \end{bmatrix}\), then ....

• A. \(a=13,\ b=-3,\ c=0\)
• B. \(a=13,\ b=3,\ c=-2\)
• C. \(a=13,\ b=-7,\ c=2\)
• D. \(a=13,\ b=-3,\ c=2\)

Hint:

Compute the determinant and the adjoint of the matrix, then compare the entries of the inverse.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
\(A^{-1}=\dfrac{\text{adj}(A)}{|A|}\).

Step 2: Find the Determinant
\(|A|=1(7-20)-2(7-10)+3(4-2)=-13+6+6=-1\).

Step 3: Find the Adjoint
The cofactors give \(\text{adj}(A)=\begin{bmatrix}-13&-2&7\\3&1&-2\\2&0&-1\end{bmatrix}\).

Step 4: Compare Entries
\(A^{-1}=-\text{adj}(A)=\begin{bmatrix}13&2&-7\\-3&-1&2\\-2&0&1\end{bmatrix}\).
Comparing with the given matrix: \(a=13\), \(c=2\), \(b=-3\).

Final Answer:
The values are \(a=13, b=-3, c=2\). \[ \boxed{\text{(D) }a=13,\ b=-3,\ c=2} \]