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Section5.2Computing Determinants

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

Task137
(Strang 5.2.2) Compute determinants of the following matrices using the big formula. Are the columns of these matrices linearly independent? \begin{equation*} A = \begin{pmatrix} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{pmatrix}, B = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}, C = \begin{pmatrix} A & 0 \\ 0 & A \end{pmatrix}, D = \begin{pmatrix} A & 0 \\ 0 & B \end{pmatrix}. \end{equation*}
Task138
(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*}
Task139
(Strang 5.2.4) Use cofactors to compute the determinants below: \begin{equation*} \det \begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 \\ 1 & 1 & 0 & 1 \\ 1 & 0 & 0 & 1 \end{pmatrix}, \qquad \det \begin{pmatrix} 1 & 0 & 0 & 2 \\ 0 & 3 & 4 & 5 \\ 5 & 4 & 0 & 3 \\ 2 & 0 & 0 & 1 \end{pmatrix}. \end{equation*}
Task140
(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 \(\det A \neq 0\).
Task141
Decide if the columns of this matrix are linearly dependent without doing any row operations: \begin{equation*} A = \begin{pmatrix} 4 & 21\\ 3 & 16 \end{pmatrix}. \end{equation*}
Task142
Complete this matrix to one with determinant zero in four genuinely different ways. How did you make that happen? \begin{equation*} X = \begin{pmatrix} 2 & 1 & \bullet \\ 1 & 1 & \bullet \\ -1 & 1 & \bullet \end{pmatrix}. \end{equation*}