📘 Question
Solve the matrix equation:
\[
\begin{bmatrix}a + 4 & 3b \\ 8 & -6\end{bmatrix}
=
\begin{bmatrix}2a + 2 & b + 2 \\ 8 & a – 8b\end{bmatrix}
\]
Find the value of \(a – 2b\).
✏️ Step-by-Step Solution
Step 1: Compare corresponding elements
- \(a + 4 = 2a + 2\)
- \(3b = b + 2\)
- \(-6 = a – 8b\)
Step 2: Solve equations
From first equation:
\[
a + 4 = 2a + 2 \Rightarrow a = 2
\]
From second equation:
\[
3b = b + 2 \Rightarrow 2b = 2 \Rightarrow b = 1
\]
Check third equation:
\[
a – 8b = 2 – 8 = -6 \quad ✔
\]
Step 3: Find required value
\[
a – 2b = 2 – 2(1)
\]
\[
= 0
\]
✅ Final Answer
\[
\boxed{0}
\]
💡 Key Concept
If two matrices are equal, then their corresponding elements must be equal. This gives a system of equations that can be solved easily.