Determinants and Invertibility

How it works

A two-by-two determinant measures signed area scaling and detects invertibility. A zero determinant means a linear transformation collapses the plane to a line or point. Divide the adjugate by the determinant only when it is nonzero.

\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc

Worked examples

Example 1

Find the determinant of $\displaystyle \begin{pmatrix}2&1\\3&4\end{pmatrix}$ .

The determinant is $\displaystyle 8-3=5$ , so the matrix is invertible.

Example 2

Is $\displaystyle \begin{pmatrix}1&2\\2&4\end{pmatrix}$ invertible?

No. Its determinant is $\displaystyle 4-4=0$ ; the rows are proportional.

Problems

Problem 4.5.1

Find the inverse of $\displaystyle \begin{pmatrix}2&0\\0&3\end{pmatrix}$ .

Show worked solution

The inverse is $\displaystyle \begin{pmatrix}1/2&0\\0&1/3\end{pmatrix}$ . Multiplication gives the identity.

Problem 4.5.2

For what k is $\displaystyle \begin{pmatrix}k&2\\3&6\end{pmatrix}$ singular?

Show worked solution

Set determinant $\displaystyle 6k-6=0$ to obtain $\displaystyle k=1$ .

Problem 4.5.3

A planar linear transformation has determinant $\displaystyle -3$ . What happens to area and orientation?

Show worked solution

Areas multiply by $\displaystyle 3$ and orientation reverses. The sign encodes orientation rather than negative physical area.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 4.5.1

Problem 4.5.2

Problem 4.5.3

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