Compute the products. Some of these may be undefined. Exercises marked \(\mathrm{T}\) should be done by using technology. The others should be done in two ways: by hand and by using technology where possible. $\left[\begin{array}{lll}1 & 3 & -1\end{array}\right]\left[\begin{array}{r}9 \\\ 1 \\ -1\end{array}\right]$

Use technology to find the inverse of the given matrix (when it exists). Round all entries in your answer to two decimal places. $$ \left[\begin{array}{rl} 3.56 & 1.23 \\ -1.01 & 0 \end{array}\right] $$

In Example 3 we said that, if a square matrix \(A\) row-reduces to a matrix with a row of zeros, then it is singular. Why?

Use row reduction to find the inverses of the given matrices if they exist, and check your answers by multiplication. $$ \left[\begin{array}{rrr} 1 & -1 & 3 \\ 0 & 1 & 3 \\ 1 & 1 & 1 \end{array}\right] $$

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 \(45^{\circ} ?\)

Make up an application whose solution reads as follows: "Total revenue $=\left[\begin{array}{lll}10 & 100 & 30\end{array}\right]\left[\begin{array}{rrr}10 & 0 & 3 \\ 1 & 2 & 0 \\ 0 & 1 & 40\end{array}\right],$