91Ó°ÊÓ

In Exercises 1 and \(2,\) find the vector \(\mathbf{x}\) determined by the given coordinate vector \([\mathbf{x}]_{\mathcal{B}}\) and the given basis \(\mathcal{B}\) . Illustrate your answer with a figure, as in the solution of Practice Problem \(2 .\) $$\mathcal{B}=\left\\{\left[\begin{array}{l}{1} \\\ {1}\end{array}\right],\left[\begin{array}{r}{2} \\\ {-1}\end{array}\right]\right\\},[\mathbf{x}]_{\mathcal{B}}=\left[\begin{array}{l}{3} \\\ {2}\end{array}\right]$$

Find formulas for X, Y, and Z in terms of A, B, and C, and justify your calculations. In some cases, you may need to make assumptions about the size of a matrix in order to produce a formula. [Hint: Compute the product on the left, and set it equal to the right side.] \(\left[\begin{array}{ll}{A} & {B} \\ {0} & {I}\end{array}\right]\left[\begin{array}{lll}{X} & {Y} & {Z} \\ {0} & {0} & {I}\end{array}\right]=\left[\begin{array}{lll}{I} & {0} & {0} \\ {0} & {0} & {I}\end{array}\right]\)

Solve the Leontief production equation for an economy with three sectors, given that $$ C=\left[\begin{array}{ccc}{.2} & {.2} & {.0} \\ {.3} & {.1} & {.3} \\ {.1} & {.0} & {.2}\end{array}\right] \text { and } \mathbf{d}=\left[\begin{array}{c}{40} \\ {60} \\ {80}\end{array}\right] $$

Can a square matrix with two identical columns be invertible? Why or why not?