1. RC Order: Always count **Rows** (Horizontal) then **Columns** (Vertical). Size $= R \times C$.
2. Entry Address: $a_{ij}$ means the entry in the $i$-th Row and $j$-th Column.
3. Identity Check: The Identity Matrix ($I$) is always Square and has 1s on the main diagonal.
4. The Transpose ($A^T$): Flip the rows into columns.
Give the size (Rows $\times$ Columns) for each matrix.
$A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$
$B = \begin{bmatrix} 10 \\ 20 \\ 30 \end{bmatrix}$
$C = \begin{bmatrix} 5 & -1 \\ 0 & 2 \end{bmatrix}$
$D = \begin{bmatrix} 1 & 0 & 0 & 7 \end{bmatrix}$
Using Matrix $M = \begin{bmatrix} 10 & 20 & 30 \\ 40 & 50 & 60 \\ 70 & 80 & 90 \end{bmatrix}$.
What is the value at address $m_{23}$?
What is the value at address $m_{31}$?
Where is the number "20" located? (Give the address $m_{ij}$).
If a matrix has 10 rows and 10 columns, how many total entries (spots) are in the grid? If you only fill the "Main Diagonal" ($i=j$), how many spots are you using? Why is the diagonal such a efficient way to organize a plan?
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The Trinity Matrix: Construct a $3 \times 3$ Identity Matrix.
The Encampment: A community is organized into 2 Rows and 2 Columns.
Row 1: {Peter, James}
Row 2: {John, Andrew}
Represent this as a Matrix of their ages (Peter=30, James=28, John=25, Andrew=27).
Given $A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$.
Task: Find the Transpose $A^T$.
(Hint: The 1st Row becomes the 1st Column).
Objective: Explain the Matrix grid to a younger sibling using a muffin tin or egg carton.
The Activity:
1. Label the rows and columns of the tin with tape.
2. Put a toy in one cup.
3. Ask: "What is the address of this toy?" (e.g., Row 1, Column 3).
The Lesson: "Everything in God's house has a 'Cup' and an 'Address'. Nothing is lost when we have a good Grid."
Response: __________________________________________________________