Practice Determinant in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The determinant is a scalar value computed from a square matrix that encodes important geometric and algebraic information. For a 2×22 \times 2 matrix [abcd]\begin{bmatrix} a & b \\ c & d \end{bmatrix}, the determinant is ad−bcad - bc. A nonzero determinant means the matrix is invertible.

The determinant measures how a matrix scales area (in 2D) or volume (in 3D). If det⁡(A)=3\det(A) = 3, the transformation described by AA triples all areas. If det⁡(A)=0\det(A) = 0, the transformation collapses space into a lower dimension (like squishing a plane into a line), which is why the matrix has no inverse.

Showing a random 20 of 50 problems.

Example 1

medium
Find the determinant of (120030004)\begin{pmatrix} 1 & 2 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{pmatrix}.

Example 2

challenge
A 2×22 \times 2 matrix AA has det⁡(A)=7\det(A) = 7. What is det⁡(A−1)\det(A^{-1}) and why?

Example 3

medium
Find kk so that [3k68]\begin{bmatrix} 3 & k \\ 6 & 8 \end{bmatrix} is singular.

Example 4

medium
If AA is 2×22 \times 2 with det⁡(A)=−5\det(A) = -5, compute det⁡(A3)\det(A^3).

Example 5

easy
Find the determinant of (3612)\begin{pmatrix} 3 & 6 \\ 1 & 2 \end{pmatrix}.

Example 6

medium
A 2×22\times 2 matrix with two identical rows has determinant ____.

Example 7

medium
If a 2×22 \times 2 matrix has det⁡=0\det = 0, what can you say about its rows?

Example 8

challenge
For 2×22 \times 2 matrices with det⁡(A)=3\det(A) = 3 and det⁡(B)=4\det(B) = 4, find det⁡(AB)\det(AB).

Example 9

easy
Find det⁡[4712]\det\begin{bmatrix} 4 & 7 \\ 1 & 2 \end{bmatrix}.

Example 10

medium
Compute det⁡[121034005]\det\begin{bmatrix} 1 & 2 & 1 \\ 0 & 3 & 4 \\ 0 & 0 & 5 \end{bmatrix}.

Example 11

easy
The determinant of a diagonal matrix [7003]\begin{bmatrix} 7 & 0 \\ 0 & 3 \end{bmatrix} equals ____.

Example 12

hard
Use cofactor expansion along column 2 to compute det⁡[302154201]\det\begin{bmatrix} 3 & 0 & 2 \\ 1 & 5 & 4 \\ 2 & 0 & 1 \end{bmatrix}.

Example 13

easy
Find the determinant of the identity (1001)\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}.

Example 14

medium
Swapping the two rows of a 2×22 \times 2 matrix changes the determinant in what way?

Example 15

easy
Find det⁡[5234]\det\begin{bmatrix} 5 & 2 \\ 3 & 4 \end{bmatrix}.

Example 16

easy
What is det⁡[0000]\det\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}?

Example 17

medium
Find the determinant of (6435)\begin{pmatrix} 6 & 4 \\ 3 & 5 \end{pmatrix}.

Example 18

easy
Compute det⁡[9463]\det\begin{bmatrix} 9 & 4 \\ 6 & 3 \end{bmatrix} and decide whether the matrix is invertible.

Example 19

medium
Compute the 3×33 \times 3 determinant of (201132011)\begin{pmatrix} 2 & 0 & 1 \\ 1 & 3 & 2 \\ 0 & 1 & 1 \end{pmatrix} by cofactor expansion along row 1.

Example 20

hard
The triangle with vertices (0,0)(0,0), (4,1)(4,1), and (2,5)(2,5) has what area?
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