1. Cross Product: Calculate the main diagonal ($a \times d$) first.
2. Subtract the Side: Subtract the opposite diagonal ($b \times c$).
3. Check for Zero: If the result is 0, the matrix has no area (it is Singular).
4. Substance Rule: A positive determinant means the plan creates space.
Calculate $\det A$ for each matrix.
$A = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}$
$B = \begin{bmatrix} 5 & -2 \\ 1 & 4 \end{bmatrix}$
$C = \begin{bmatrix} 1 & 10 \\ 0 & 5 \end{bmatrix}$
Which of these plans has zero substance? Write "Singular" or "Non-Singular."
$\begin{bmatrix} 2 & 4 \\ 4 & 8 \end{bmatrix}$
$\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$
$\begin{bmatrix} 10 & 5 \\ 20 & 10 \end{bmatrix}$
If you have a matrix where the second row is all zeros $\begin{bmatrix} a & b \\ 0 & 0 \end{bmatrix}$, what is the determinant? Why does it make sense that a plan with a "Dead Row" has zero area?
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Find the value of $x$ that makes the determinant of $\begin{bmatrix} x & 4 \\ 2 & 8 \end{bmatrix}$ equal to 0.
1. Find the determinant of the $2 \times 2$ Identity matrix $I$.
2. Find the determinant of $10 \cdot I$.
Question: Is the determinant of $10 \cdot I$ equal to 10? Or is it 100? Why does scaling Both rows multiply the area by the square of the scalar?
Objective: Explain Determinants to a younger sibling using a book and a tabletop.
The Activity:
1. Open a book slightly. "See the space inside? That's an Area."
2. Close the book completely. "Is there any space left?"
3. "In math, the Determinant tells us if the book is open or closed."
The Lesson: "God wants our life to be an 'Open Book' where there is plenty of room for Him. We keep it open by being different and interesting!"
Response: ___________________________________________________________