Question

Q. If every element of a square non singular matrix A is multiplied by k and the new matrix is denoted by B then |A^{-1}| and |B^{-1}| are related as

• A. |A^{-1}| = k|B^{-1}|
• B. |A^{-1}| = \frac{1}{k}|B^{-1}|
• C. |A^{-1}| = k^{n}|B^{-1}|
• D. |A^{-1}| = k^{-n}|B^{-1}|

Answer 1

Answer: (C)

|A^{-1}| = k^{n}|B^{-1}|

Answer 2

Step 1: Let A be an n×n nonsingular matrix.
Step 2: Define B = kA, so
$\det(B)=\det(kA)=k^n\det(A).$
Step 3: Then

$$B^{-1}=(kA)^{-1}=\tfrac{1}{k}A^{-1} \quad\Longrightarrow\quad \det(B^{-1})=\bigl(\tfrac{1}{k}\bigr)^n\det\bigl(A^{-1}\bigr).$$

Step 4: Rearranging gives

$$\det(A^{-1})=k^n\det(B^{-1}),$$

hence $|A^{-1}|=k^n|B^{-1}|$.