Question

Raya sent an encoded message back to Darnell that provides the location of the next clue.C=[77,9, 75, -35, 41, -13, 80,-40, 44, -2, 28, -14, 127, -1]To decode the message, Darnell will need to use the inverse of the 2 x 2 encoding matrix, A. Complete the following problems to determine the next clue. 1.)What is the determinant of A? Show your work or explain. (3 points)A=[3 4 1 -2 2.)What is the inverse matrix of A? Show how to get A–1. (4 points)3.)What is A–1 × C? (4 points)4.)Describe how to use the resultant matrix from part c to decode the message. What does the decoded message say?

Answer 1

1. The matrix A is:

`A=\begin{bmatrix}{3} & {4} & {} \\ {1} & {-2} & {} \\ {} & {} & {}\end{bmatrix}`

The determinant of a 2x2 matrix can be found as follows:

`\begin{gathered} \det \begin{bmatrix}{a} & {b} & {} \\ {c} & {d} & {} \\ {} & {} & {}\end{bmatrix}=a* d-b* c \\ \text{Thus }\det |A|=3*(-2)-4*1 \\ \det |A|=-6-4 \\ \det |A|=-10 \end{gathered}`

The determinant of matrix A is -10

2. The inverse of a matrix is given by:

`\begin{gathered} A^(-1)=(1)/(\det |A|)\begin{bmatrix}{d} & {-b} & {} \\ {-c} & {a} & {} \\ {} & {} & {}\end{bmatrix} \\ A^(-1)=(1)/(-10)\begin{bmatrix}{-2} & {-4} & {} \\ {-1} & {3} & {} \\ {} & {} & {}\end{bmatrix} \\ A^(-1)=\begin{bmatrix}{(-2)/(-10)} & {(-4)/(-10)} & {} \\ {(-1)/(-10)} & {(3)/(-10)} & {} \\ {} & {} & {}\end{bmatrix}=\begin{bmatrix}{(1)/(5)} & {(2)/(5)} & {} \\ {(1)/(10)} & {-(3)/(10)} & {} \\ {} & {} & {}\end{bmatrix} \end{gathered}`