(" + m11 + " * " + m22 + ") – (" + m12 + " * " + m21 + ") = " + det;}else{var part1 = m11 * (m22 * m33 – m23 * m32);var part2 = m12 * (m21 * m33 – m23 * m31);var part3 = m13 * (m21 * m32 – m22 * m31);det = part1 – part2 + part3;steps = "Expansion by Row 1:
" + m11 + " * (" + m22 + "*" + m33 + " – " + m23 + "*" + m32 + ")
– " + m12 + " * (" + m21 + "*" + m33 + " – " + m23 + "*" + m31 + ")
+ " + m13 + " * (" + m21 + "*" + m32 + " – " + m22 + "*" + m31 + ")
= " + det;}document.getElementById('resultValue').innerHTML = det;if(document.getElementById('steps').checked){document.getElementById('stepDetails').innerHTML = steps;document.getElementById('stepDetails').style.display = 'block';}else{document.getElementById('stepDetails').style.display = 'none';}}
Calculator Use
This determinant calculator is a specialized tool used to find the determinant of square matrices, specifically for 2×2 and 3×3 dimensions. The determinant is a scalar value that provides critical information about a matrix, such as whether it is invertible or represents a system of linear equations with a unique solution.
To use this tool, simply select the matrix dimensions from the dropdown, enter the values for each element (a11 through a33), and click "Calculate". You can also enable the "Show expansion details" checkbox to see the step-by-step arithmetic used in the Laplace expansion method.
- Matrix Size
- Choose between a 2×2 matrix (4 elements) or a 3×3 matrix (9 elements).
- Elements (aij)
- The numerical values of the matrix where 'i' represents the row and 'j' represents the column.
- Expansion Details
- A breakdown showing the calculation of minors and cofactors used to reach the final determinant.
How It Works
The method for calculating a determinant depends on the order of the matrix. For a 2×2 matrix, the process is straightforward subtraction of cross-products. For a 3×3 matrix, our calculator uses the cofactor expansion along the first row.
2×2 Formula: det(A) = ad – bc
3×3 Formula: det(A) = a(ei – fh) – b(di – fg) + c(dh – eg)
- 2×2 Matrix: If A = [[a, b], [c, d]], the determinant is the difference between the product of the main diagonal and the anti-diagonal.
- 3×3 Matrix: This follows the Laplace expansion where we multiply the elements of the first row by the determinants of their corresponding 2×2 sub-matrices (minors).
- Singularity: If the result is 0, the matrix is "singular," meaning it has no inverse.
Calculation Example
Example: Find the determinant of a 3×3 matrix with the following values: Row 1 [1, 2, 3], Row 2 [0, 4, 5], Row 3 [1, 0, 6].
Step-by-step solution:
- Identify elements: a=1, b=2, c=3, d=0, e=4, f=5, g=1, h=0, i=6
- Apply expansion: 1*(4*6 – 5*0) – 2*(0*6 – 5*1) + 3*(0*0 – 4*1)
- Simplify parts: 1*(24) – 2*(-5) + 3*(-4)
- Combine: 24 + 10 – 12
- Result = 22
Common Questions
Can this determinant calculator handle 4×4 matrices?
Currently, this calculator supports 2×2 and 3×3 matrices. 4×4 matrices require significantly more complex expansion or row reduction methods like Gaussian elimination.
What does a negative determinant mean?
A negative determinant indicates that the linear transformation associated with the matrix involves a change in "orientation" or a "reflection" in space. In terms of area or volume, the absolute value is used.
Why is the determinant 0 for some matrices?
If a determinant is zero, it means the matrix is linearly dependent. This happens if one row is a multiple of another, or if there is a row of all zeros. In geometry, this means the transformation collapses the space into a lower dimension.