📘 Question
If
\[
A =
\begin{bmatrix}
1 & -1 \\
2 & -1
\end{bmatrix},
\quad
B =
\begin{bmatrix}
a & 1 \\
b & -1
\end{bmatrix}
\]
and
\[
(A + B)^2 = A^2 + B^2
\]
Find the values of \(a\) and \(b\).
✏️ Step-by-Step Solution
Step 1: Use identity
\[ (A+B)^2 = A^2 + AB + BA + B^2 \]
Given:
\[
A^2 + AB + BA + B^2 = A^2 + B^2
\]
So,
\[
AB + BA = 0
\]
Step 2: Compute \(AB\)
\[
AB =
\begin{bmatrix}
1 & -1 \\
2 & -1
\end{bmatrix}
\begin{bmatrix}
a & 1 \\
b & -1
\end{bmatrix}
=
\begin{bmatrix}
a – b & 2 \\
2a – b & 3
\end{bmatrix}
\]
Step 3: Compute \(BA\)
\[
BA =
\begin{bmatrix}
a & 1 \\
b & -1
\end{bmatrix}
\begin{bmatrix}
1 & -1 \\
2 & -1
\end{bmatrix}
=
\begin{bmatrix}
a + 2 & -a -1 \\
b – 2 & -b +1
\end{bmatrix}
\]
Step 4: Use \(AB + BA = 0\)
\[
AB + BA =
\begin{bmatrix}
2a – b + 2 & 1 – a \\
2a + 2b – 2 & 4 – b
\end{bmatrix}
=
0
\]
Step 5: Solve equations
- \(1 – a = 0 \Rightarrow a = 1\)
- \(4 – b = 0 \Rightarrow b = 4\)
Check consistency ✔
✅ Final Answer
\[
\boxed{a = 1,\quad b = 4}
\]
💡 Key Concept
If \((A+B)^2 = A^2 + B^2\), then:
\[
AB + BA = 0
\]
Use this condition to solve unknown matrix elements.