Matrix

Determinant of Matrix

N/A

Calculation Steps

Matrix Determinant Calculator (tool above) is a GUI tool to calculate the determinant of a matrix. It can calculate determinant for 2x2, 3x3 and 4x4 matrices.

This tool also shows the step-by-step calculation of determinant.

It is tested for matrices up to 4x4. For larger matrices, the calculation takes a little longer. If you have any feedback or suggestions to improve this tool, please do not hesitate to contact us.

Determinant is a single or scalar value that can be calculated from a square matrix (matrix having uniform number of rows and columns).

Non-square matrices do not have a determinant. Therefore, non-square matrices do not have inverse either.

To use the above matrix determinant calculator, select the dimension using select box

Enter the matrix values in input boxes

Click on Calculate button to get the determinant

Matrix determinant don't have a clear purpose directly. However, it is used in many operations and concepts in linear algebra

Determinant is used to calculate inverse of a matrix

Determinant of matrix is used to calculate the inverse of a matrix

It is also used to solve system of linear equations using Cramer's rule

Non-square matrices don't have determinant because determinant operation is only defined for a square matrix

Non-square matrices don't have invertible property, and therefore don't have determinant as well

Both eigen values are determinants are concepts related to scalar values of square matrix in linear algebra and they both serve different purpose in linear algebra.

Determinants are used to calculate existence of inverse of a matrix, and determinant is a volume scaling factor (when applied to transformation)

And Eigen values are used to calculate how matrix scales a vector, when linear transformation is applied

Read more about Eigen Values & Eigen Vectors, Determinants

For example there is a 2x2 matrix [ [a, b], [c, d] ]

The determinant is calculated by multiplying principal diagonal elements and subtracting the product of other diagonal elements

```
\(A = \begin{bmatrix}
a & b \\
c & d
\end{bmatrix}
\)
```

```
\(\text{det}(A) = ad - bc
\)
```

Nice and simple

For example there is a 3x3 matrix

```
\(A = \begin{bmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{bmatrix}
\)
```

Step 1: Divide the original matrix into 2x2 matrix by removing rows and columns one by one

After removing first row and first column, we get a 2x2 matrix

```
\(A = \begin{bmatrix}
e & f \\
h & i
\end{bmatrix}
\)
```

After removing second row and second column, we get a 2x2 matrix

```
\(B = \begin{bmatrix}
d & f \\
g & i
\end{bmatrix}
\)
```

After removing third row and third column, we get a 2x2 matrix

```
\(C = \begin{bmatrix}
d & e \\
g & h
\end{bmatrix}
\)
```

Step 2: Calculate the determinant of each 2x2 matrix

```
\(det(A) = a(ei - fh)
det(B) = -b(di - fg)
det(C) = c(dh - eg)
\)
```

Here I calculated the determinants of 2x2 matrices A, B, and C and also multiplied them with the corresponding elements of the original matrix

But as you can see the determinant (B) is negative, because its the second element of first row. Every second element is multiplied with - sign.

Step 3: Add the results of step 2

```
\(det(A) + det(B) + det(C)
\)
```

```
\(= a(ei - fh) - b(di - fg) + c(dh - eg)
\)
```

This is the determinant of 3x3 matrix.

If you find it difficult to calculate the determinant of 3x3 matrix, you can use the above matrix determinant calculator to calculate it easily.

You can also refer to this video from Khan Academy for better understanding

**Tip**: Choose the row or columns with the highest number of zeros to make the calculation easier (so that you can remove more elements)

For example there is a 4x4 matrix

```
\(A = \begin{bmatrix}
a & b & c & d \\
e & f & g & h \\
i & j & k & l \\
m & n & o & p
\end{bmatrix}
\)
```

Step 1: Divide the original matrix into 3x3 matrix by removing rows and columns one by one

After removing first row and first column, we get a 3x3 matrix

```
\(A = \begin{bmatrix}
f & g & h \\
j & k & l \\
n & o & p
\end{bmatrix}
\)
```

```
\(det(A) = f(kp - lo) - g(jp - ln) + h(jo - kn)
\)
```

After removing second row and second column, we get a 3x3 matrix

```
\(B = \begin{bmatrix}
e & g & h \\
i & k & l \\
m & o & p
\end{bmatrix}
\)
```

```
\(det(B) = -e(kp - lo) + g(ip - lm) - h(io - km)
\)
```

After removing third row and third column, we get a 3x3 matrix

```
\(C = \begin{bmatrix}
e & f & h \\
i & j & l \\
m & n & p
\end{bmatrix}
\)
```

```
\(det(C) = e(jp - ln) - f(ip - lm) + h(in - jm)
\)
```

After removing fourth row and fourth column, we get a 3x3 matrix

```
\(D = \begin{bmatrix}
e & f & g \\
i & j & k \\
m & n & o
\end{bmatrix}
\)
```

```
\(det(D) = -e(jo - kn) + f(io - km) - g(in - jm)
\)
```

Step 2: Calculate the determinant of each 3x3 matrix

Step 3: Add the results of step 2

```
\(det(A) + det(B) + det(C) + det(D)
\)
```

```
\(
= f(kp - lo) - g(jp - ln) + h(jo - kn) - e(kp - lo) + g(ip - lm) - h(io - km) + e(jp - ln) - f(ip - lm) + h(in - jm) - e(jo - kn) + f(io - km) - g(in - jm)
\)
```

simplify the above equation

```
\(= a(fkxop - glxjn + hlyjn - ejxop) - b(ekxop - glxim + hlxim - djxop) + c(ekxjn - fkxim + hlxg - djxg) - d(ejxjn - fkxim + glxg - djxf)
\)
```

This is the determinant of 4x4 matrix.

This is a long and complex formula and its not feasible to remember this as well. You have to calculate it step by step by understanding the process instead of memorizing the formula.

Its called calculating determinants by elimination method, where you eliminate rows and columns one by one to get the determinant of a matrix.

Because finding determinants is lengthy process, I have built this matrix determinant calculator to make it easy for you to calculate the determinant of a matrix.

However, if you wish to understand the process here's a video from The Organic Chemistry Tuto for better understanding of calculating determinants of 4x4 matrix

For a 2x2 matrix, of element [ [a, b], [c, d] ] The determinant is calculated as ad - bc Calculating 2x2 matrix determinant is simple and easy

For a 3x3 matrix, of element [ [a, b, c], [d, e, f], [g, h, i] ]

The determinant is calculated as a(ei - fh) - b(di - fg) + c(dh - eg)

For a 4x4 matrix, of element [ [a, b, c, d], [e, f, g, h], [i, j, k, l], [m, n, o, p] ]

The determinant is calculated as

a(fkxop - glxjn + hlyjn - ejxop) - b(ekxop - glxim + hlxim - djxop) + c(ekxjn - fkxim + hlxg - djxg) - d(ejxjn - fkxim + glxg - djxf)

As the dimension increase, the calculation of determinant becomes more complex and time consuming

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