Matrix Determinant Calculator

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About the Matrix Determinant Calculator

The determinant calculator calculates the determinant of a square matrix of order 10\hspace{0.2em} 10 \hspace{0.2em} and below.

Usage Guide

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i. Valid Inputs

The size of the square matrix can be any positive integer from 1\hspace{0.2em} 1 \hspace{0.2em} to 10\hspace{0.2em} 10 \hspace{0.2em}.

Each element in the matrix can be a real number in any format — integers, decimals, fractions, or even mixed numbers. Here are a few examples.

  • Whole numbers or decimals → 2\hspace{0.2em} 2 \hspace{0.2em}, 4.25\hspace{0.2em} -4.25 \hspace{0.2em}, 0\hspace{0.2em} 0 \hspace{0.2em}, 0.33\hspace{0.2em} 0.33 \hspace{0.2em}
  • Fractions → 2/3\hspace{0.2em} 2/3 \hspace{0.2em}, 1/5\hspace{0.2em} -1/5 \hspace{0.2em}
  • Mixed numbers → 51/4\hspace{0.2em} 5 \hspace{0.5em} 1/4 \hspace{0.2em}

ii. Example

If you would like to see an example of the calculator's working, just click the "example" button.

iii. Share

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By checking the "include calculation" checkbox, you can share your calculation as well.

Here's a quick overview of the concept of determinants.

What Is a Determinant?

A determinant is a unique scalar value associated with a square matrix and is a function of the elements of the matrix. It is common denoted as det(A)\hspace{0.2em} \text{det} (A) \hspace{0.2em} or A\hspace{0.2em} |A| \hspace{0.2em}, A\hspace{0.2em} A \hspace{0.2em} being the relevant matrix.

Determinant of a 2×2\hspace{0.2em} 2 \times 2 \hspace{0.2em} Matrix

By definition, the determinant of a 2×2\hspace{0.2em} 2 \times 2 \hspace{0.2em} matrix is the difference between the products of the elements along the main diagonal ↘ and those along the antidiagonal ↙.

Let me explain. Consider the matrix below.

A=[abcd]A \hspace{0.25em} = \hspace{0.25em} \begin{bmatrix} a & b\\ c & d \end{bmatrix}

The determinant for the matrix above would be

A=adbc|A| = ad - bc

Determinants of Larger Matrices

Before extending the idea of determinants to matrices of a higher order, it's helpful to understand what we mean by the terms "minor" and "cofactor" mean.

Minors and Cofactors

Consider the determinant below. aij\hspace{0.2em} a_{ij} \hspace{0.2em} represents the element in the ith\hspace{0.2em} i^{\text{th}} \hspace{0.2em} row and jth\hspace{0.2em} j^{\text{th}} \hspace{0.2em} column.

a11a12a13a1na21a22a23a2na31a32a33a3nan1an2an3ann \begin{vmatrix} a_{11} & a_{12} & a_{13} & \cdots & a_{1n} \\ a_{21} & a_{22} & a_{23} & \cdots & a_{2n} \\ a_{31} & a_{32} & a_{33} & \cdots & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & a_{n3} & \cdots & a_{nn} \\ \end{vmatrix}

Now the minor corresponding to the element aij\hspace{0.2em} a_{ij} \hspace{0.2em} is the smaller determinant obtained by omitting the the ith\hspace{0.2em} i^{\text{th}} \hspace{0.2em} row and jth\hspace{0.2em} j^{\text{th}} \hspace{0.2em} column of the original matrix.

So the minor corresponding to the element a22\hspace{0.2em} a_{22} \hspace{0.2em} (denoted by M22\hspace{0.2em} M_{22} \hspace{0.2em}) would be —

M22=a11a13a1na31a33a3nan1an3annM_{22} \hspace{0.25em} = \hspace{0.25em} \begin{vmatrix} a_{11} & a_{13} & \cdots & a_{1n} \\ a_{31} & a_{33} & \cdots & a_{3n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n3} & \cdots & a_{nn} \\ \end{vmatrix}

Cofactor of an element aij\hspace{0.2em} a_{ij} \hspace{0.2em} is defined as —

Cij=(1)(i+j)MijC_{ij} \hspace{0.25em} = \hspace{0.25em} (-1)^{(i + j)} \cdot M_{ij}

So the cofactor of a22\hspace{0.2em} a_{22} \hspace{0.2em} would be

C22=(1)(2+2)M22=M22\begin{align*} C_{22} \hspace{0.25em} &= \hspace{0.25em} (-1)^{(2 + 2)} \cdot M_{22} \\[1em] &= \hspace{0.25em} M_{22} \end{align*}

Laplace Expansion

To calculate the value of a determinant larger than 2×2\hspace{0.2em} 2 \times 2 \hspace{0.2em}, we pick one row (or column) and add the products of each element in that row (or column) with its cofactor.

So for a square matrix of order n\hspace{0.2em} n \hspace{0.2em}, the determinant is given by the formula

A=j=1naijCij|A| \hspace{0.25em} = \hspace{0.25em} \sum_{j = 1}^{n} a_{ij} \hspace{0.15em} C_{ij}

Here, i\hspace{0.2em} i \hspace{0.2em} can take integral value from 1\hspace{0.2em} 1 \hspace{0.2em} to n\hspace{0.2em} n \hspace{0.2em}.

This method is known as Laplace expansion along one particular row or column.

Determinant Calculation — Example


Calculate the determinant of the matrix below.

A=[724150392]A \hspace{0.25em} = \hspace{0.25em} \begin{bmatrix} 7 & 2 & 4 \\ -1 & 5 & 0\\ 3 & 9 & -2 \end{bmatrix}


To calculate the determinant of this 3×3\hspace{0.2em} 3 \times 3 \hspace{0.2em} matrix, let's go with Laplace expansion along the second row. Why we choose the second row will make sense in a moment.

So let's calculate the cofactors for each of the elements in the second row. We start with the cofactor of a21\hspace{0.2em} a_{21} \hspace{0.2em}.

C21=(1)(2+1)2492=1(2×2+4×9)=32\begin{align*} C_{21} \hspace{0.25em} &= \hspace{0.25em} (-1)^{(2 + 1)} \cdot \begin{vmatrix} 2 & 4 \\ 9 & -2 \end{vmatrix} \\[1.5em] &= \hspace{0.25em} -1 \cdot (2 \times -2 + 4 \times 9) \\[1.5em] &= \hspace{0.25em} -32 \end{align*}

Similarly, the cofactor of a22\hspace{0.2em} a_{22} \hspace{0.2em}

C22=(1)(2+2)7432=1(7×2+4×3)=2\begin{align*} C_{22} \hspace{0.25em} &= \hspace{0.25em} (-1)^{(2 + 2)} \cdot \begin{vmatrix} 7 & 4 \\ 3 & -2 \end{vmatrix} \\[1.5em] &= \hspace{0.25em} 1 \cdot (7 \times -2 + 4 \times 3) \\[1.5em] &= \hspace{0.25em} -2 \end{align*}

We don't need to calculate C23\hspace{0.2em} C_{23} \hspace{0.2em} because as you'll see, it will get multiplied with 0\hspace{0.2em} 0 \hspace{0.2em} and hence won't matter. and that's precisely why we selected the second row for expansion — less calculation.

Now, as we learned earlier to calculate the determinant, we add the products of each element in the row (or column) with its cofactor. So,

A=a21C21+a22C22+a23C23=1(32)+5(2)+0C23=32+10+0=22\begin{align*} |A| \hspace{0.25em} &= \hspace{0.25em} a_{21} \cdot C_{21} + a_{22} \cdot C_{22} + a_{23} \cdot C_{23} \\[1em] &= \hspace{0.25em} -1 \cdot (-32) + 5 \cdot (-2) + 0 \cdot C_{23} \\[1em] &= \hspace{0.25em} 32 + -10 + 0 \\[1em] &= \hspace{0.25em} 22 \end{align*}

And that's it.

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