Check One-One / Onto
🎥 Video Explanation
📝 Question
Let \(M\) be the set of all \(2 \times 2\) matrices with real entries.
Function: \[ f : M \to \mathbb{R}, \quad f(A)=|A| \]
- A. one-one and onto
- B. neither one-one nor onto
- C. one-one not one-one
- D. onto but not one-one
✅ Solution
🔹 Step 1: Understanding Function
For a matrix: \[ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]
\[ |A| = ad – bc \]
—🔹 Step 2: Check Injective
Different matrices can have same determinant.
Example:
\[ \begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}, \quad \begin{pmatrix}2 & 0 \\ 0 & \tfrac{1}{2}\end{pmatrix} \]
Both have determinant = 1
❌ Not one-one
—🔹 Step 3: Check Surjective
For any real number \(k\), choose:
\[ A = \begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix} \]
\[ |A| = k \]
Every real value possible ⇒ ✔️ Onto
—🔹 Final Answer
\[ \boxed{\text{Option D: onto but not one-one}} \]