Study guide and
15 practice problems
on:
Determinant of a 3x3 matrix
$\newcommand{\bfA}{\mathbf{A}}$ $\newcommand{\bfB}{\mathbf{B}}$ $\newcommand{\bfC}{\mathbf{C}}$ $\newcommand{\bfF}{\mathbf{F}}$ $\newcommand{\bfI}{\mathbf{I}}$ $\newcommand{\bfa}{\mathbf{a}}$ $\newcommand{\bfb}{\mathbf{b}}$ $\newcommand{\bfc}{\mathbf{c}}$ $\newcommand{\bfd}{\mathbf{d}}$ $\newcommand{\bfe}{\mathbf{e}}$ $\newcommand{\bfi}{\mathbf{i}}$ $\newcommand{\bfj}{\mathbf{j}}$ $\newcommand{\bfk}{\mathbf{k}}$ $\newcommand{\bfn}{\mathbf{n}}$ $\newcommand{\bfr}{\mathbf{r}}$ $\newcommand{\bfu}{\mathbf{u}}$ $\newcommand{\bfv}{\mathbf{v}}$ $\newcommand{\bfw}{\mathbf{w}}$ $\newcommand{\bfx}{\mathbf{x}}$ $\newcommand{\bfy}{\mathbf{y}}$ $\newcommand{\bfz}{\mathbf{z}}$
Study Guide
The method of diagonals for computing the determinant of a 3x3 matrix
(2 problems)
Related topics
The method of diagonals for computing the determinant of a 3x3 matrix
(2 problems)
Determinants
(23 problems)
Multivariable calculus
(147 problems)
Practice problems
Show that $\bfa \times \bfb$ is perpendicular to $\bfa$ by computing a dot product.
Solution
Compute $\bfi \times (\bfi + \bfk)$ in two ways:
By the determinant formula
By expanding the sum and recalling the cross products of standard coordinate vectors with each other
Solution
What is the relationship of $\bfa \times \bfb$ and $\bfb \times \bfa$? Prove it using components.
Solution
Use the method of diagonals to compute the determinant $$\text{det } \begin{pmatrix} 0 & 1 & 2 \\ 3 & -1 & 0 \\ 1 & -2 & 1 \end{pmatrix}$$
Solution
Compute the determinant of $$\begin{pmatrix}1 & -1& 1 \\ 1 & 1 & 1\\1 & 2 & 4 \end{pmatrix}$$
Solution
For a $2 \times 2$ matrix, if one row is a multiple of the other, the determinant is zero. Use this fact and the method of minors and cofactors to show that the determinant of a $3 \times 3$ matrix is zero if one row is a multiple of another.
Solution
Compute the determinant $$\text{det } \begin{pmatrix} 1 & 5 & 0 \\ 2 & 1 & 0 \\ 1 & 0 & 3 \end{pmatrix}$$ by minors and cofactors along the second column.
Solution
Compute the determinant $$\text{det } \begin{pmatrix} 1 & 5 & 0 \\ 2 & 1 & 0 \\ 1 & 0 & 3 \end{pmatrix}$$ by minors and cofactors along the first row.
Solution
Find a unit vector perpendicular to $\langle 1, 1, 1\rangle$ and $\langle 1, 0, 1 \rangle.$
Solution
Consider the triangle in three-space given by $\langle a, 0, 0\rangle, \langle 0, b, 0 \rangle, \langle 0, 0, c \rangle.$ Find a vector that is perpendicular to the triangle and has length equal to the area of the triangle.
Solution
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