Study guide and
16 practice problems
on:
Determinant of a 2x2 matrix
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The determinant of a $2 \times 2$ matrix is $$
\text{det } \begin{pmatrix}a&b\\c&d\end{pmatrix} = ad-bc$$
Related topics
Determinants
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Multivariable calculus
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Practice problems
Show that $\bfa \times \bfb$ is perpendicular to $\bfa$ by computing a dot product.
Solution
Compute the determinant $$\text{det } \begin{pmatrix} 0 & 1 \\ 2 & 3 \end{pmatrix}$$
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
Show that if the rows of a $2 \times 2$ matrix are multiples of each other, then the determinant of the matrix is zero.
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
Sketch the parallelogram spanned by $\langle 1,1\rangle$ and $\langle 0, 1\rangle$. Use a cross product to find its area.
Solution
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