Section 1
Determinants by Cofactor Expansion
Find all the minors and cofactors of the ma$\operatorname{trix} A$.$$A=\left[\begin{array}{rrr}1 & -2 & 3 \\6 & 7 & -1 \\-3 & 1 & 4\end{array}\right]$$
Find all the minors and cofactors of the ma$\operatorname{trix} A$.$$A=\left[\begin{array}{lll}1 & 1 & 2 \\3 & 3 & 6 \\0 & 1 & 4\end{array}\right]$$
Let$$A=\left[\begin{array}{rrrr}4 & -1 & 1 & 6 \\0 & 0 & -3 & 3 \\4 & 1 & 0 & 14 \\4 & 1 & 3 & 2\end{array}\right]$$
Let$$A=\left[\begin{array}{rrrr}2 & 3 & -1 & 1 \\-3 & 2 & 0 & 3 \\3 & -2 & 1 & 0 \\3 & -2 & 1 & 4\end{array}\right]$$Find(a) $M_{32}$ and $C_{32}$(b) $M_{44}$ and $C_{44}$(c) $M_{41}$ and $C_{41}$(d) $M_{24}$ and $C_{24}$
Evaluate the determinant of the given matrix. If the matrix is invertible, use Fquation (2) to find its inverse.$$\left[\begin{array}{rr}3 & 5 \\-2 & 4\end{array}\right]$$
Evaluate the determinant of the given matrix. If the matrix is invertible, use Fquation (2) to find its inverse.$$\left[\begin{array}{ll}4 & 1 \\8 & 2\end{array}\right]$$
Evaluate the determinant of the given matrix. If the matrix is invertible, use Fquation (2) to find its inverse.$$\left[\begin{array}{rr}-5 & 7 \\-7 & -2\end{array}\right]$$
Evaluate the determinant of the given matrix. If the matrix is invertible, use Fquation (2) to find its inverse.$$\left[\begin{array}{cc}\sqrt{2} & \sqrt{6} \\4 & \sqrt{3}\end{array}\right]$$
Use the arrow technique to evaluate the determinant.$$\left|\begin{array}{cc}a-3 & 5 \\-3 & a-2\end{array}\right|$$
Use the arrow technique to evaluate the determinant.$$\left|\begin{array}{rrr}-2 & 7 & 6 \\5 & 1 & -2 \\3 & 8 & 4\end{array}\right|$$
Use the arrow technique to evaluate the determinant.$$\left|\begin{array}{rrr}-2 & 1 & 4 \\3 & 5 & -7 \\1 & 6 & 2\end{array}\right|$$
Use the arrow technique to evaluate the determinant.$$\left|\begin{array}{rrr}-1 & 1 & 2 \\3 & 0 & -5 \\1 & 7 & 2\end{array}\right|$$
Use the arrow technique to evaluate the determinant.$$\left|\begin{array}{rrr}3 & 0 & 0 \\2 & -1 & 5 \\1 & 9 & -4\end{array}\right|$$
Use the arrow technique to evaluate the determinant.$$\left|\begin{array}{ccc}c & -4 & 3 \\2 & 1 & c^{2} \\4 & c-1 & 2\end{array}\right|$$
Find all values of $\lambda$ for which $\operatorname{det}(A)=0$.$$A=\left[\begin{array}{cc}\lambda-2 & 1 \\-5 & \lambda+4\end{array}\right]$$
Find all values of $\lambda$ for which $\operatorname{det}(A)=0$.$$A=\left[\begin{array}{ccc}\lambda-4 & 0 & 0 \\0 & \lambda & 2 \\0 & 3 & \lambda-1\end{array}\right]$$
Find all values of $\lambda$ for which $\operatorname{det}(A)=0$.$$A=\left[\begin{array}{cc}\lambda-1 & 0 \\2 & \lambda+1\end{array}\right]$$
Find all values of $\lambda$ for which $\operatorname{det}(A)=0$.$$A=\left[\begin{array}{ccc}\lambda-4 & 4 & 0 \\-1 & \lambda & 0 \\0 & 0 & \lambda-5\end{array}\right]$$
Evaluate the determinant in Exercise 13 by a cofactor expansion along(a) the first row.(b) the first column.(c) the sccond row.(d) the sccond column.(c) the third row.(f) the third column.
- Evaluate the determinant in Exercise 12 by a cofactor expansion along(a) the first row.(b) the first column.(c) the second row.(d) the second column.(c) the third row.(f) the third column.
Evaluate det $(A)$ by a cofactor expansion along a row or column of your choice.$$4=\left[\begin{array}{rrr}-3 & 0 & 7 \\2 & 5 & 1 \\-1 & 0 & 5\end{array}\right]$$
Evaluate det $(A)$ by a cofactor expansion along a row or column of your choice.$$A=\left[\begin{array}{rrr}3 & 3 & 1 \\1 & 0 & -4 \\1 & -3 & 5\end{array}\right]$$
Evaluate det $(A)$ by a cofactor expansion along a row or column of your choice.$$A=\left[\begin{array}{lll}1 & k & k^{2} \\1 & k & k^{2} \\1 & k & k^{2}\end{array}\right]$$
Evaluate det $(A)$ by a cofactor expansion along a row or column of your choice.$$A=\left[\begin{array}{ccc}k+1 & k-1 & 7 \\2 & k-3 & 4 \\5 & k+1 & k\end{array}\right]$$
Evaluate det $(A)$ by a cofactor expansion along a row or column of your choice.$$A=\left[\begin{array}{rrrr}3 & 3 & 0 & 5 \\2 & 2 & 0 & -2 \\4 & 1 & -3 & 0 \\2 & 10 & 3 & 2\end{array}\right]$$
Evaluate det $(A)$ by a cofactor expansion along a row or column of your choice.$$A=\left[\begin{array}{lllrl}4 & 0 & 0 & 1 & 0 \\3 & 3 & 3 & -1 & 0 \\1 & 2 & 4 & 2 & 3 \\9 & 4 & 6 & 2 & 3 \\2 & 2 & 4 & 2 & 3\end{array}\right]$$
Evaluate the determinant of the given matrix by inspection.$$\left[\begin{array}{ccc}1 & 0 & 0 \\0 & -1 & 0 \\0 & 0 & 1\end{array}\right]$$
Evaluate the determinant of the given matrix by inspection.$$\left[\begin{array}{lll}2 & 0 & 0 \\0 & 2 & 0 \\0 & 0 & 2\end{array}\right]$$
Evaluate the determinant of the given matrix by inspection.$$\left[\begin{array}{llll}0 & 0 & 0 & 0 \\1 & 2 & 0 & 0 \\0 & 4 & 3 & 0 \\1 & 2 & 3 & 8\end{array}\right]$$
Evaluate the determinant of the given matrix by inspection.$$\left[\begin{array}{llll}1 & 1 & 1 & 1 \\0 & 2 & 2 & 2 \\0 & 0 & 3 & 3 \\0 & 0 & 0 & 4\end{array}\right]$$
Evaluate the determinant of the given matrix by inspection.$$\left[\begin{array}{cccr}1 & 2 & 7 & -3 \\0 & 1 & -4 & 1 \\0 & 0 & 2 & 7 \\0 & 0 & 0 & 3\end{array}\right]$$
Evaluate the determinant of the given matrix by inspection.$$\left[\begin{array}{rrrr}-3 & 0 & 0 & 0 \\1 & 2 & 0 & 0 \\40 & 10 & -1 & 0 \\100 & 200 & -23 & 3\end{array}\right]$$
In each part, show that the value of the determinant is independent of $\theta$(a) $\left|\begin{array}{rc}\sin \theta & \cos \theta \\ -\cos \theta & \sin \theta\end{array}\right|$(b) $\left|\begin{array}{ccc}\sin \theta & \cos \theta & 0 \\ -\cos \theta & \sin \theta & 0 \\ \sin \theta-\cos \theta & \sin \theta+\cos \theta & 1\end{array}\right|$
Show that the matrices$$A=\left[\begin{array}{ll}a & b \\0 & c\end{array}\right] \quad \text { and } \quad B=\left[\begin{array}{ll}d & e \\0 & f\end{array}\right]$$commute if and only if$$\left|\begin{array}{ll}b & a-c \\e & d-f\end{array}\right|=0$$
By inspection, what is the relationship between the following determinants?$$d_{1}=\begin{array}{lll}a & b & c \\d & 1 & f \\g & 0 & 1\end{array} | \text { and } d_{2}=\left|\begin{array}{ccc}a+\lambda & b & c \\d & 1 & f \\g & 0 & 1\end{array}\right|$$
Show that$$\operatorname{det}(A)=\frac{1}{2} \operatorname{tr}(A) \quad 1$$for every $2 \times 2$ matrix $A$
What can you say about an $n$ th-order determinant all of whose entries are $1 ?$ Explain.
What is the maximum number of zeros that a $3 \times 3$ matrix can have without having a zero determinant? Explain.
Explain why the determinant of a matrix with integer entrics must be an integer.
Prove that $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),$ and $\left(x_{3}, y_{3}\right)$ are collinear points if and only if$$\left|\begin{array}{lll}x_{1} & y_{1} & 1 \\x_{2} & y_{2} & 1 \\x_{3} & y_{3} & 1\end{array}\right|=0$$
Prove that the equation of the line through the distinct points $\left(a_{1}, b_{1}\right)$ and $\left(a_{2}, b_{2}\right)$ can be writtcn as$$\left|\begin{array}{lll}x & y & 1 \\a_{1} & b_{1} & 1 \\a_{2} & b_{2} & 1\end{array}\right|=0$$
Prove that if $A$ is upper triangular and $B_{i j}$ is the matrix that results when the $i$ th row and $j$ th column of $A$ are deleted, then $B_{i j}$ is upper triangular if $i<j$