1. Suppose a triangle in the plane has an area of 1 square inch. What is the area of the image of the triangle under the linear transformation determined by the matrix with first row (1,2) and second row (3,4)?
Answer: 2 square inches. This is the absolute value of the product of the area of the triangle (1) with with the determinant of the matrix (-2). Note that the matrix reverses the orientation of the triangle.
2. What is the volume of the image of the unit cube under the linear transformation with first row (1, 0, 1), second row (1,1,1) and final row (0,0,0)?
Answer: Zero volume. This is the product of the volume of the cube (1) with the determinant of the matrix (0). This makes sense because the range of the matrix has dimension 2, so the image of the cube is in a plane and thus has 0 volume.
3. What is the determinant of the matrix with columns (1,2,3), (4,5,6), and (7,8,9)?
Answer: 0.
4. What is the determinant of the matrix with rows (1,1,1), (1,2,3) and (2,2,5)?
Answer: 3.