You are given a technology matrix A and an external demand vector \(D\). Find the corresponding production vector \(X .\) [HINT: See Quick Example \(1 .]\) $$ A=\left[\begin{array}{lll} 0.5 & 0.1 & 0 \\ 0 & 0.5 & 0.1 \\ 0 & 0 & 0.5 \end{array}\right], D=\left[\begin{array}{l} 3,000 \\ 3,800 \\ 2,000 \end{array}\right] $$

Use technology to find the inverse of the given matrix (when it exists). Round all entries in your answer to two decimal places. $$ \left[\begin{array}{clcc} 0.01 & 0.32 & 0 & 0.04 \\ -0.01 & 0 & 0 & 0.34 \\ 0 & 0.32 & -0.23 & 0.23 \\ 0 & 0.41 & 0 & 0.01 \end{array}\right] $$

Compute the determinant of the given matrix. If the determinant is nonzero, use the formula for inverting \(a 2 \times 2\) matrix to calculate the inverse of the given matrix. $$ \left[\begin{array}{ll} 4 & 1 \\ 0 & 2 \end{array}\right] $$

Rotations If a point \((x, y)\) in the plane is rotated counterclockwise about the origin through an angle of \(45^{\circ},\) its new coordinates \(\left(x^{\prime}, y^{\prime}\right)\) are given by $$ \left[\begin{array}{l} x^{\prime} \\ y^{\prime} \end{array}\right]=R\left[\begin{array}{l} x \\ y \end{array}\right] $$ where \(R\) is the \(2 \times 2\) matrix $\left[\begin{array}{rr}a & -a \\ a & a\end{array}\right]\( and \)a=\sqrt{1 / 2} \approx$ a. If the point (2,3) is rotated counterclockwise through an angle of \(45^{\circ},\) what are its (approximate) new coordinates? b. Multiplication by what matrix would result in a counterclockwise rotation of \(90^{\circ} ? 135^{\circ} ?\) (Express the matrices in terms of \(R\).) [HINT: Think of a rotation through \(90^{\circ}\) as two successive rotations through \(45^{\circ} .\) c. Multiplication by what matrix would result in a clockwise rotation of \(45^{\circ} ?\)

Give an example of two matrices \(A\) and \(B\) such that \(A B\) is defined but $B A$ is not defined.

Rotations If a point \((x, y)\) in the plane is rotated counterclockwise about the origin through an angle of \(60^{\circ},\) its new coordinates \(\left(x^{\prime}, y^{\prime}\right)\) are given by $$ \left[\begin{array}{l} x^{\prime} \\ y^{\prime} \end{array}\right]=S\left[\begin{array}{l} x \\ y \end{array}\right] $$ where \(S\) is the \(2 \times 2\) matrix $\left[\begin{array}{rr}a & -b \\ b & a\end{array}\right]\( and \)a=1 / 2$ and \(b=\sqrt{3 / 4} \approx 0.8660 .\) a. If the point (2,3) is rotated counterclockwise through an angle of \(60^{\circ},\) what are its (approximate) new coordinates? b. Referring to Exercise \(61,\) multiplication by what matrix would result in a counterclockwise rotation of \(105^{\circ} ?\) (Express the matrices in terms of \(S\) and the matrix \(R\) from Exercise 61.) [HINT: Think of a rotation through \(105^{\circ}\) as a rotation through \(60^{\circ}\) followed by a rotation through \(\left.45^{\circ} .\right]\) c. Multiplication by what matrix would result in a clockwise rotation of \(60^{\circ}\) ?