Exercise 231Linear algebra › DeterminantsHigh schoolAdded 3 Nov 2026

Determinant of a 2×2 matrix

#determinant#2x2 matrices#invertibility

Statement

Compute the determinant of each matrix and decide whether it is invertible:

(a) $A = \begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix}$ ;
(b) $B = \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}$ ;
(c) $C = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$ .

Hint

For a $2\times2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ , the determinant is $ad-bc$ . The matrix is invertible iff the determinant is nonzero.

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Solution

(a) $\det A = 3\cdot2 - 1\cdot5 = 6-5=1$ . Invertible.

(b) $\det B = 1\cdot4 - 2\cdot2 = 0$ . Not invertible.

(c) $\det C = \cos\theta\cdot\cos\theta - (-\sin\theta)(\sin\theta) = \cos^2\theta+\sin^2\theta = 1$ . Invertible for all $\theta$ (it is a rotation matrix).

Determinant of a 2×2 matrix

Statement

Solution

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