# SubsectionThe Assignment

• Read Chapter 5 section 2 of Strang.
• Read the following and complete the exercises below.

# SubsectionLearning Goals

Before class, a student should be able to compute the determinant by using cofactors. A student should also be able to compute a determiant using the “big formula” for matrices of size 2 or 3.

Some time after class, a student should be comfortable with the different parts of the invertible matrix theorem.

# SubsectionDiscussion: The Importance of the Determinant

Strang devotes all of his energy in this section to the different ways to compute the determinant. I don't have much to add to that.

The real importance of the determinant is described in the following theorem. Note that this is a special result for square matrices. The shape is crucial for this result.

# SubsectionExercises

(Strang 5.2.3) Show that $\det(A)=0$, no matter what values are used to fill in the five unknowns marked with dots. What are the cofactors of row 1? What is the rank of $A$? What are the six terms in the big formula? \begin{equation*} A = \begin{pmatrix} \bullet & \bullet & \bullet \\ 0 & 0 & \bullet \\ 0 & 0 & \bullet \end{pmatrix}. \end{equation*}
(Strang 5.2.5) What is the smallest arrangement of zeros you can place in a $4 \times 4$ matrix to guarantee that its determinant is zero? Try to place as many non-zero entries as you can while keeping $\det A \neq 0$.