Q.If $A = \begin{bmatrix}1 & 2\\ 4 & 1\end{bmatrix}$, find $A^2 + 2A + 7I$.

Concept: Matrix Addition — we compute $A^2$ first, then scale $A$ and $I$, and add the results entrywise.

Step 1: Compute $A^2$.

Step 2: Compute $2A$ and $7I$.

$2A = \begin{bmatrix}2 & 4\\8 & 2\end{bmatrix}, \quad 7I = \begin{bmatrix}7 & 0\\0 & 7\end{bmatrix}$ …

Step 1 — $A^2=A\cdot A$.

Step 2 — $2A$ and $7I$.

$2A=\begin{bmatrix}2 & 4\\8 & 2\end{bmatrix},\qquad 7I=\begin{bmatrix}7 & 0\\0 & 7\end{bmatrix}.$ …

Method: Evaluating a matrix polynomial

To evaluate an expression such as $A^2+2A+7I$, compute each term as a matrix and add them entrywise. The only subtlety is that the constant term is a scalar multiple of the identity, not a scalar added to every entry.

Steps

Step 1: Compute $A^2$ by matrix multiplication.

Each entry of $A^2$ is a row of $A$ dotted with a column of $A$.

Step 2: Compute each scalar-times-matrix term.

For $2A$, multiply every entry of $A$ by $2$. …

Common Mistakes

Mistake 1: Treating $7I$ as "add $7$ to every entry".

Why it's wrong: $7I=\begin{bmatrix}7&0\\0&7\end{bmatrix}$, so $7$ is added only to the diagonal entries; adding it to the off-diagonal entries is wrong. Correct approach: build $7I$ explicitly before adding.

Mistake 2: Computing $A^2$ as the entrywise square of $A$.

Why it's wrong: $A^2$ means the matrix product $A\cdot A$, not squaring each entry. Correct approach: use row-by-column multiplication. …