Determinant Calculator

determinant calculator

2 x 2 Matrix3 x 3 Matrix

Answer:

Determinant (det A) =

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Calculator Use

The determinant calculator is a specialized tool designed to help students, engineers, and mathematicians quickly find the determinant of a square matrix. Whether you are working with a 2×2 or a 3×3 matrix, this tool automates the tedious manual calculations required in linear algebra.

To use this calculator, simply select your matrix dimensions, input the numerical values for each element (a11, a12, etc.), and click "Calculate." You can also check the "Show Solution Steps" box to see the underlying arithmetic used to reach the final answer.

Matrix Size

Choose between a 2×2 (two rows, two columns) or a 3×3 (three rows, three columns) matrix.

Elements (a11, a12, etc.)

These represent the individual numbers located at specific row and column positions within the matrix.

Show Solution Steps

When enabled, the tool provides the breakdown of the formula and intermediate calculations.

How It Works

The determinant is a scalar value that provides critical information about a square matrix. For instance, if the determinant is zero, the matrix is "singular" and does not have an inverse. The formulas used by our determinant calculator are as follows:

2×2 Determinant Formula

det(A) = (a * d) – (b * c)

Where 'a' and 'd' are the main diagonal elements, and 'b' and 'c' are the off-diagonal elements.

3×3 Determinant Formula (Laplace Expansion)

det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)

• a, b, c: Elements of the first row.
• (ei – fh): The determinant of the 2×2 minor matrix remaining after removing the first row and first column.
• (di – fg): The determinant of the minor matrix after removing the first row and second column.
• (dh – eg): The determinant of the minor matrix after removing the first row and third column.

Calculation Example

Example: Find the determinant of a 3×3 matrix where the values are Row 1: [1, 2, 3], Row 2: [4, 5, 6], and Row 3: [7, 8, 9].

Step-by-step solution:

1. Identify elements: a=1, b=2, c=3, d=4, e=5, f=6, g=7, h=8, i=9.
2. Apply Laplace expansion: 1(5*9 – 6*8) – 2(4*9 – 6*7) + 3(4*8 – 5*7)
3. Calculate minors: 1(45 – 48) – 2(36 – 42) + 3(32 – 35)
4. Simplify: 1(-3) – 2(-6) + 3(-3)
5. Final addition: -3 + 12 – 9 = 0
6. Result: det(A) = 0 (This matrix is singular).

Common Questions

What does a determinant of zero mean?

A determinant of zero indicates that the matrix is singular or non-invertible. In terms of linear transformations, it means the matrix collapses the space into a lower dimension (e.g., a 3D volume becomes a 2D plane or 1D line).

Can a determinant be negative?

Yes, a determinant can be any real number, including negative values. A negative determinant usually indicates a change in the orientation of the vector space after the transformation is applied.

Is this calculator suitable for 4×4 matrices?

This specific tool is optimized for 2×2 and 3×3 matrices, which cover the vast majority of academic and practical requirements. For 4×4 or higher, the expansion methods become significantly more complex, involving more layers of minor calculations.