1. Dimension Match: You can only add or subtract matrices if they are the Same Size.
2. Entry Addition: Add numbers that share the same address ($a_{ij} + b_{ij}$).
3. Scalar Distribution: Multiply every entry inside the brackets by the scalar outside.
4. Sign Steward: Watch out for double negatives (e.g., $5 - (-3) = 8$).
Perform the indicated operation. If it is impossible, write "Dimension Mismatch."
$\begin{bmatrix} 1 & 5 \\ 3 & 0 \end{bmatrix} + \begin{bmatrix} 2 & -1 \\ 4 & 7 \end{bmatrix}$
$\begin{bmatrix} 10 & 20 \end{bmatrix} + \begin{bmatrix} 5 \\ 5 \end{bmatrix}$
$\begin{bmatrix} 5 & 2 & 1 \\ 0 & -3 & 4 \end{bmatrix} - \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}$
If you add two $2 \times 2$ matrices, what are the dimensions of the result? Why is it impossible to gain a "New Row" just by adding? What does this tell you about the Stability of the Plan?
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Multiply the matrix by the scalar.
$5 \cdot \begin{bmatrix} 2 & 0 \\ -1 & 4 \end{bmatrix}$
$-2 \cdot \begin{bmatrix} 3 & 10 \\ 5 & -2 \end{bmatrix}$
$0.5 \cdot \begin{bmatrix} 10 & 20 & 30 \\ 40 & 50 & 60 \end{bmatrix}$
If $A = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & -1 \\ 2 & 3 \end{bmatrix}$...
Find $2A + B$.
Three tribes report their grain harvests in a $3 \times 1$ matrix: $\mathbf{H} = \begin{bmatrix} 100 \\ 200 \\ 150 \end{bmatrix}$.
1. A drought causes every harvest to decrease by 20 units. (Subtract a $3 \times 1$ constant matrix).
2. Then, a miracle blessing multiplies the remaining harvest by 3.
Task: Find the final matrix of the grain.
Objective: Explain Matrix Addition to a younger student using Lego blocks.
The Activity:
1. Give them a 2x2 grid of blocks.
2. Have them add another 2x2 grid on top.
3. Ask: "How many blocks high is the corner at Row 1, Column 1?"
The Lesson: "Addition means making the grid Stronger, not bigger. We keep everyone in their same spot and just add more grace."
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