Calculation Results
Determinant Calculator With Steps (2x2, 3x3, n×n)
Use this determinant calculator with steps to compute determinants of 2x2, 3x3, and larger matrices instantly with explanations.
Determinant Calculator for Different Matrix Sizes
- 2x2 determinant calculator
- 3x3 determinant with steps
- n×n determinant solver
A determinant is a scalar value that represents certain properties of a square matrix, such as invertibility, area/volume scaling, and linear independence of rows/columns.
A determinant is written as:
\[ \det(A) \quad \text{or} \quad |A| \]
2. Determinant Formulas
2.1 Determinant of a 2×2 Matrix
\[ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \qquad \det(A) = ad - bc \]
2.2 Determinant of a 3×3 Matrix
Using Sarrus' rule: \[ A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]
\[ \det(A) = aei + bfg + cdh - ceg - bdi - afh \]
3. Solved Examples
Example 1 – 2×2 Determinant
Compute: \[ A = \begin{pmatrix} 3 & 5 \\ 2 & 4 \end{pmatrix} \]
\[ \det(A) = 3 \cdot 4 - 5 \cdot 2 = 12 - 10 = 2 \]
Example 2 – 3×3 Determinant
Compute: \[ B = \begin{pmatrix} 1 & 2 & 3 \\ 0 & -1 & 4 \\ 2 & 1 & 0 \end{pmatrix} \]
Using the formula:
\[ \det(B) = 1(-1 \cdot 0 - 4 \cdot 1) - 2(0 \cdot 0 - 4 \cdot 2) + 3(0 \cdot 1 - (-1)\cdot 2) \]
\[ = 1(-4) - 2(-8) + 3(2) = -4 + 16 + 6 = 18 \]
4. Common Mistakes When Computing Determinants
- Mixing up signs when expanding (especially in 3×3 determinants).
- Forgetting negative numbers during multiplication.
- Applying the 2×2 formula incorrectly.
- Trying to compute determinants of non-square matrices (impossible).
- Not realizing row/column swaps change the sign of the determinant.
5. Practice Problems (With Collapsible Answers)
Problem 1
\[ \begin{pmatrix} 4 & 1 \\ 3 & 2 \end{pmatrix} \]
Show Answer
\[ 4 \cdot 2 - 1 \cdot 3 = 5 \]Problem 2
\[ \begin{pmatrix} 2 & 3 & 1 \\ 0 & -1 & 4 \\ 1 & 2 & 0 \end{pmatrix} \]
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\[ \det = -2 \]Problem 3
\[ \begin{pmatrix} 7 & 2 \\ 5 & -3 \end{pmatrix} \]
Show Answer
\[ 7(-3) - 2 \cdot 5 = -31 \]Problem 4
\[ \begin{pmatrix} 3 & 0 & 2 \\ 1 & 4 & 5 \\ 2 & -1 & 3 \end{pmatrix} \]
Show Answer
\[ \det = 29 \]Frequently Asked Questions (FAQ)
Q: How do you calculate the determinant of a matrix?
The determinant can be calculated using formulas for small matrices or expansion and row operations for larger ones.
Q: Does this determinant calculator show steps?
Yes. It provides step-by-step calculations for better understanding.
The determinant is closely connected with the RREF calculator, matrix rank, and eigenvalues of a matrix.
Learn matrix multiplication more easily in just 2 minutes with this free game.
