Section 5.2 Computing Determinants
Subsection 5.2.1 The Assignment
- Read Chapter 5 section 2 of Strang.
- Read the following and complete the exercises below.
Subsection 5.2.2 Learning 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.
Subsection 5.2.3 Discussion: 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.
Theorem 5.2.1.
(The Invertible Matrix Theorem)Let \(A\) be an \(n\times n\) matrix. Then the following conditions are equivalent:
- The columns of \(A\) are linearly independent.
- The columns of \(A\) are a spanning set for \(\mathbb{R}^n\text{.}\)
- The colums of \(A\) are a basis for \(\mathbb{R}^n\text{.}\)
- The rows of \(A\) are linearly independent.
- The rows of \(A\) are a spanning set for \(\mathbb{R}^n\text{.}\)
- The rows of \(A\) are a basis for \(\mathbb{R}^n\text{.}\)
- For any choice of vector \(b \in \mathbb{R}^n\text{,}\) the system of linear equations \(Ax = b\) has a unique solution.
- \(A\) is invertible.
- The transpose \(A^T\) is invertible.
- \(\det(A) \neq 0\text{.}\)
- \(\det(A^T) \neq 0\text{.}\)
Subsection 5.2.4 Exercises
(a)
(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*}
(b)
(Strang 5.2.3) Show that \(\det(A)=0\text{,}\) 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 \(A\text{?}\) 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*}
(c)
(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*}
(d)
(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 \(\det A \neq 0\text{.}\)(e)
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*}
(f)
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*}