91Ó°ÊÓ

The columns of matrix \(T\) show the coordinates of the vertices of a triangle. Matrix \(A\) is a transformation matrix. $$A=\left[\begin{array}{rr} 0 & -1 \\ 1 & 0 \end{array}\right], \quad T=\left[\begin{array}{lll} 1 & 2 & 3 \\ 1 & 4 & 2 \end{array}\right]$$ (a) Find \(A T\) and \(A A T .\) Then sketch the original triangle and the two transformed triangles. What transformation does \(A\) represent? (b) A triangle is determined by \(A A T .\) Describe the transformation process that produces the triangle determined by \(A T\) and then the triangle determined by \(T\)

A fruit grower raises two crops, apples and peaches. Each of these crops is shipped to three different outlets. The number of units of crop \(i\) that are shipped to outlet \(j\) is represented by \(a_{i j}\) in the matrix $$A=\left[\begin{array}{lll} 125 & 100 & 75 \\ 100 & 175 & 125 \end{array}\right]$$ The profit per unit is represented by the matrix \\[ B=\left[\begin{array}{ll} \$ 3.50 & \$ 6.00 \end{array}\right] \\] Find the product \(B A\) and state what each entry of the product represents.

Express the column matrix b as a linear combination of the columns of \(A\) $$A=\left[\begin{array}{rrr} 1 & 1 & -5 \\ 1 & 0 & -1 \\ 2 & -1 & -1 \end{array}\right], \quad \mathbf{b}=\left[\begin{array}{l} 3 \\ 1 \\ 0 \end{array}\right]$$

Express the column matrix b as a linear combination of the columns of \(A\) $$A=\left[\begin{array}{rr} -3 & 5 \\ 3 & 4 \\ 4 & -8 \end{array}\right], \quad \mathbf{b}=\left[\begin{array}{r} -22 \\ 4 \\ 32 \end{array}\right]$$