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Section 2.1 Exercises

Let

$A = [\begin{matrix} 3 & 2 & 4 \\ 1 & - 2 & 3 \\ 2 & 3 & 2 \end{matrix}]$ $A = [\begin{matrix} 3 & 2 & 4 \\ 1 & - 2 & 3 \\ 2 & 3 & 2 \end{matrix}]$
1. Find the values of $det (M_{21})$ $det (M_{21})$ , $det (M_{22})$ $det (M_{22})$ , and $det (M_{23})$ $det (M_{23})$ .
2. Find the values of $A_{21}$ $A_{21}$ , $A_{22}$ $A_{22}$ , and $A_{23}$ $A_{23}$ .
3. Use your answers from part (b) to compute det(A).
Use determinants to determine whether the following $2 \times 2$ $2 \times 2$ matrices are nonsingular:
1. $[\begin{matrix} 3 & 5 \\ 2 & 4 \end{matrix}]$ $[\begin{matrix} 3 & 5 \\ 2 & 4 \end{matrix}]$
2. $[\begin{matrix} 3 & 6 \\ 2 & 4 \end{matrix}]$ $[\begin{matrix} 3 & 6 \\ 2 & 4 \end{matrix}]$
3. $[\begin{matrix} 3 & - 6 \\ 2 & 4 \end{matrix}]$ $[\begin{matrix} 3 & - 6 \\ 2 & 4 \end{matrix}]$
Evaluate the following determinants:
1. $| \begin{matrix} 3 & 5 \\ - 2 & - 3 \end{matrix} |$ $| \begin{matrix} 3 & 5 \\ - 2 & - 3 \end{matrix} |$
2. $| \begin{matrix} 5 & - 2 \\ - 8 & 4 \end{matrix} |$ $| \begin{matrix} 5 & - 2 \\ - 8 & 4 \end{matrix} |$
3. $| \begin{matrix} 3 & 1 & 2 \\ 2 & 4 & 5 \\ 2 & 4 & 5 \end{matrix} |$ $| \begin{matrix} 3 & 1 & 2 \\ 2 & 4 & 5 \\ 2 & 4 & 5 \end{matrix} |$
4. $| \begin{matrix} 4 & 3 & 0 \\ 3 & 1 & 2 \\ 5 & - 1 & - 4 \end{matrix} |$ $| \begin{matrix} 4 & 3 & 0 \\ 3 & 1 & 2 \\ 5 & - 1 & - 4 \end{matrix} |$
5. $| \begin{matrix} 1 & 3 & 2 \\ 4 & 1 & - 2 \\ 2 & 1 & 3 \end{matrix} |$ $| \begin{matrix} 1 & 3 & 2 \\ 4 & 1 & - 2 \\ 2 & 1 & 3 \end{matrix} |$
6. $| \begin{matrix} 2 & - 1 & 2 \\ 1 & 3 & 2 \\ 5 & 1 & 6 \end{matrix} |$ $| \begin{matrix} 2 & - 1 & 2 \\ 1 & 3 & 2 \\ 5 & 1 & 6 \end{matrix} |$
7. $| \begin{matrix} 2 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 6 & 2 & 0 \\ 1 & 1 & - 2 & 3 \end{matrix} |$ $| \begin{matrix} 2 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 6 & 2 & 0 \\ 1 & 1 & - 2 & 3 \end{matrix} |$
8. $| \begin{matrix} 2 & 1 & 2 & 1 \\ 3 & 0 & 1 & 1 \\ - 1 & 2 & - 2 & 1 \\ - 3 & 2 & 3 & 1 \end{matrix} |$ $| \begin{matrix} 2 & 1 & 2 & 1 \\ 3 & 0 & 1 & 1 \\ - 1 & 2 & - 2 & 1 \\ - 3 & 2 & 3 & 1 \end{matrix} |$
Evaluate the following determinants by inspection:
1. $| \begin{matrix} 3 & 5 \\ 2 & 4 \end{matrix} |$ $| \begin{matrix} 3 & 5 \\ 2 & 4 \end{matrix} |$
2. $| \begin{matrix} 2 & 0 & 0 \\ 4 & 1 & 0 \\ 7 & 3 & - 2 \end{matrix} |$ $| \begin{matrix} 2 & 0 & 0 \\ 4 & 1 & 0 \\ 7 & 3 & - 2 \end{matrix} |$
3. $| \begin{matrix} 3 & 0 & 0 \\ 2 & 1 & 1 \\ 1 & 2 & 2 \end{matrix} |$ $| \begin{matrix} 3 & 0 & 0 \\ 2 & 1 & 1 \\ 1 & 2 & 2 \end{matrix} |$
4. $| \begin{matrix} 4 & 0 & 2 & 1 \\ 5 & 0 & 4 & 2 \\ 2 & 0 & 3 & 4 \\ 1 & 0 & 2 & 3 \end{matrix} |$ $| \begin{matrix} 4 & 0 & 2 & 1 \\ 5 & 0 & 4 & 2 \\ 2 & 0 & 3 & 4 \\ 1 & 0 & 2 & 3 \end{matrix} |$
Evaluate the following determinant. Write your answer as a polynomial in x:

$| \begin{matrix} a - x & b & c \\ 1 & - x & 0 \\ 0 & 1 & - x \end{matrix} |$ $| \begin{matrix} a - x & b & c \\ 1 & - x & 0 \\ 0 & 1 & - x \end{matrix} |$
Find all values of λ for which the following determinant will equal 0:

$| \begin{matrix} 2 - λ & 4 \\ 3 & 3 - λ \end{matrix} |$ $| \begin{matrix} 2 - λ & 4 \\ 3 & 3 - λ \end{matrix} |$
Let A be a $3 \times 3$ $3 \times 3$ matrix with $a_{11} = 0$ $a_{11} = 0$ and $a_{21} \neq 0$ $a_{21} \neq 0$ . Show that A is row equivalent to I if and only if

$\begin{array}{l} - a_{12} a_{21} a_{33} + a_{12} a_{31} a_{23} \\ + a_{13} a_{21} a_{32} - a_{13} a_{31} a_{22} \neq 0 \end{array}$ $\begin{array}{l} - a_{12} a_{21} a_{33} + a_{12} a_{31} a_{23} \\ + a_{13} a_{21} a_{32} - a_{13} a_{31} a_{22} \neq 0 \end{array}$
Write out the details of the proof of Theorem 2.1.3.
Prove that if a row or a column of an $n \times n$ $n \times n$ matrix A consists entirely of zeros, then $det (A) = 0$ $det (A) = 0$ .
Use mathematical induction to prove that if A is an $(n + 1) \times (n + 1)$ $(n + 1) \times (n + 1)$ matrix with two identical rows, then $det (A) = 0$ $det (A) = 0$ .
Let A and B be $2 \times 2$ $2 \times 2$ matrices.
1. Does det $(A + B) = det (A) + det (B) ?$ $(A + B) = det (A) + det (B) ?$
2. Does det $(A B) = det (A) det (B) ?$ $(A B) = det (A) det (B) ?$
3. Does det $(A B) = det (B A) ?$ $(A B) = det (B A) ?$
Justify your answers.
Let A and B be $2 \times 2$ $2 \times 2$ matrices and let

$\begin{matrix} \begin{matrix} \begin{matrix} C = [\begin{matrix} a_{11} & a_{12} \\ b_{21} & b_{22} \end{matrix}], \end{matrix} & D = [\begin{matrix} b_{11} & b_{12} \\ a_{21} & a_{22} \end{matrix}], \end{matrix} \\ E = [\begin{matrix} 0 & α \\ β & 0 \end{matrix}] \end{matrix}$ $\begin{matrix} \begin{matrix} \begin{matrix} C = [\begin{matrix} a_{11} & a_{12} \\ b_{21} & b_{22} \end{matrix}], \end{matrix} & D = [\begin{matrix} b_{11} & b_{12} \\ a_{21} & a_{22} \end{matrix}], \end{matrix} \\ E = [\begin{matrix} 0 & α \\ β & 0 \end{matrix}] \end{matrix}$
1. Show that $det (A + B) = det (A) + det (B) + det (C) + det (D)$ $det (A + B) = det (A) + det (B) + det (C) + det (D)$ .
2. Show that if $B = E A$ $B = E A$ , then $det (A + B) = det (A) + det (B)$ $det (A + B) = det (A) + det (B)$ .
Let A be a symmetric tridiagonal matrix (i.e., A is symmetric and $a_{ij} = 0$ $a_{ij} = 0$ whenever $| i - j | > 1$ $| i - j | > 1$ ). Let B be the matrix formed from A by deleting the first two rows and columns. Show that

$det (A) = a_{11} det (M_{11}) - a_{12}^{2} det (B)$ $det (A) = a_{11} det (M_{11}) - a_{12}^{2} det (B)$

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Table of Contents for Section 2.1 Exercises

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Table of Contents for
Section 2.1 Exercises