📘 Question
If \[ A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & -1 \end{bmatrix} \] find \(A^2\).
✏️ Solution (Single-Step Matrix Multiplication)
\[
A^2 =
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
a & b & -1
\end{bmatrix}
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
a & b & -1
\end{bmatrix}
\]
\[
=
\begin{bmatrix}
(1)(1)+(0)(0)+(0)(a) & (1)(0)+(0)(1)+(0)(b) & (1)(0)+(0)(0)+(0)(-1) \\
(0)(1)+(1)(0)+(0)(a) & (0)(0)+(1)(1)+(0)(b) & (0)(0)+(1)(0)+(0)(-1) \\
(a)(1)+(b)(0)+(-1)(a) & (a)(0)+(b)(1)+(-1)(b) & (a)(0)+(b)(0)+(-1)(-1)
\end{bmatrix}
\]
\[
=
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}
\]
✅ Final Answer
\[
\boxed{
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}
}
\]
💡 Key Concept
Writing full multiplication in one matrix step helps avoid mistakes and clearly shows how terms cancel out.