Chapter 6

We have seen that determinants can be used to solve linear equations, give areas of triangles in rectangular coordinates, and determine equations of lines. Not impressed with these applications? Members of the group should research an application of determinants that they find intriguing. The group should then present a seminar to the class about this application.

Write each system in the form \(A X=B\). Then solve the system by entering \(A\) and \(B\) into your graphing utility and computing \(A^{-1} B\). $$ \begin{aligned} w+x+y+z &=4 \\ w+3 x-2 y+2 z &=7 \\ 2 w+2 x+y+z &=3 \\ w-x+2 y+3 z &=5 \end{aligned} $$

Use a coding matrix A of your choice. Use a graphing utility to find the multiplicative inverse of your coding matrix. Write a cryptogram for each message. Check your result by decoding the cryptogram. Use your graphing utility to perform all necessary matrix multiplications. $$ \begin{aligned} &\begin{array}{lll} \text { A } \mathrm{R} & \mathrm{T} & \text { - E } \mathrm{N} \text { R I } \mathrm{C} \mathrm{H} \text { E } \mathrm{S} \end{array}\\\ &1\quad18\quad20\quad0\quad5\quad14\quad18\quad9\quad3\quad8\quad5\quad19 \end{aligned} $$

Which one of the following is true? a. Some nonsquare matrices have inverses. b. All square \(2 \times 2\) matrices have inverses because there is a formula for finding these inverses. \- c. Two \(2 \times 2\) invertible matrices can have a matrix sum that is not invertible. -d. To solve the matrix equation \(A X=B\) for \(X,\) multiply \(A\) and the inverse of \(B\)

Which one of the following is true? \(B,\) and \(A B\) are a. \((A B)^{-1}=A^{-1} B^{-1},\) assuming \(A\) invertible. b. \((A+B)^{-1}=A^{-1}+B^{-1},\) assuming \(A, B,\) and \(A+B\) are invertible. c. \(\left[\begin{array}{rr}1 & -3 \\ -1 & 3\end{array}\right]\) is an invertible matrix. d. None of the above is true.

Give an example of a \(2 \times 2\) matrix that is its own inverse.

If \(A=\left[\begin{array}{ll}3 & 5 \\ 2 & 4\end{array}\right],\) find \(\left(A^{-1}\right)^{-1}\)

Find values of \(a\) for which the following matrix is not invertible: $$ \left[\begin{array}{rr} 1 & a+1 \\ a-2 & 4 \end{array}\right] $$