📘 Question
Solve the matrix equation:
\[
\begin{bmatrix}xy & 4 \\ z + 6 & x + y\end{bmatrix}
=
\begin{bmatrix}8 & w \\ 0 & 6\end{bmatrix}
\]
Find the value of \(x + y + z\).
✏️ Step-by-Step Solution
Step 1: Compare corresponding elements
- \(xy = 8\)
- \(4 = w\)
- \(z + 6 = 0 \Rightarrow z = -6\)
- \(x + y = 6\)
Step 2: Solve for \(x\) and \(y\)
We have:
\[
x + y = 6 \quad \text{and} \quad xy = 8
\]
These correspond to a quadratic:
\[
t^2 – 6t + 8 = 0
\]
\[
(t – 2)(t – 4) = 0
\]
So, \(x, y = 2, 4\) (in any order).
Step 3: Compute \(x + y + z\)
\[
x + y + z = 6 + (-6)
\]
\[
= 0
\]
✅ Final Answer
\[
\boxed{0}
\]
💡 Key Concept
In matrix equality, corresponding elements are equal. This gives algebraic equations which can be solved simultaneously.