1. Determinant Check: Calculate $D = ad - bc$ first. If $D=0$, there is no inverse.
2. Swap & Flip: Swap $a$ and $d$. Flip the signs of $b$ and $c$.
3. Rescale: Multiply the entire matrix by $1/D$.
4. The Proof: Multiply $A$ by $A^{-1}$. You must get the Identity $\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$.
Find the inverse matrix $A^{-1}$ for each given matrix $A$.
$A = \begin{bmatrix} 2 & 1 \ 5 & 3 \end{bmatrix}$
$A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$
$A = \begin{bmatrix} 5 & 0 \ 0 & 5 \end{bmatrix}$ (A Scaling Matrix).
Check if the inverse exists. Write "Singular" (No Inverse) or "Non-Singular."
$\begin{bmatrix} 1 & 2 \ 2 & 4 \end{bmatrix}$
$\begin{bmatrix} 10 & 1 \ 0 & 1 \end{bmatrix}$
Take Matrix $A$ from Problem 1 and multiply it by your $A^{-1}$. Show the work for the first row. Do you get $\langle 1, 0 \rangle$? Why is the Identity the "Final Rest" of the multiplication?
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The Distorted Truth: A person's "Identity Vector" was transformed by $A = \begin{bmatrix} 1 & 1 \ 0 & 1 \end{bmatrix}$ into the result $\mathbf{b} = \begin{bmatrix} 10 \ 4 \end{bmatrix}$.
1. Find $A^{-1}$.
2. Find the original identity $\mathbf{x}$.
A secret message is hidden in the vector $\mathbf{b} = \langle 14, 2 \rangle$.
The encoding matrix was $E = \begin{bmatrix} 3 & 1 \ 1 & 1 \end{bmatrix}$.
Task: Find the decoding matrix $E^{-1}$ and find the original message $\mathbf{x}$.
(Hint: The entries of $\mathbf{x}$ are the ranks of letters in the alphabet - e.g., 6=F, 1=A).
Objective: Explain the Inverse Matrix to a younger student using a pile of sorted socks.
The Activity:
1. Scramble a pair of socks ($A$).
2. Ask: "Can we get the pair back?"
3. Have them fold the socks together ($A^{-1}$).
4. "The fold is the inverse of the scramble."
The Lesson: "Math has a special Key for every scramble. No matter how messy things look, God has a Way Back."
Response: ___________________________________________________________