Determinant Calculator

The determinant calculator simplifies the process of finding the determinants for matrices of order up to 5×5 size. Select the size of the matrix and put either real or complex numbers to evaluate their determinant matrix with the calculations for each step.

What is a Determinant?

It is a scalar values that is obtained from the elements of the square matrix. It has certain properties of the linear transformation and measures how much a linear transformation indicated by the matrix stretches. The determinant of a matrix is positive or negative depend on whether linear transformation preserves or reverses the orientation of a vector space. It is denoted as det (A), det A, or |A|.

How to Calculate the Determinant of a Matrix?

The determinant of the matrices can be calculated from the different methods but the determinant calculator computes the determinant of a 2x2, 3x3, 4x4 or higher-order square matrix. The calculator takes the complexity out of matrix calculations, making it simple and easy to find determinants for matrices of any size. In simple manually, it is calculated by multiplying its main diagonal members & reducing matrix to row echelon form. Here we give the detailed formulas for different order of matrix to find the determinant from different methods:

For 2x2 Matrix Multiplications:

No matter, which method you selected for the calculations, the determinant of matrix A = (aij)2×2 is determined by the following formula:

\( det A = \begin{vmatrix} a & b \\ c & d \end{vmatrix} \\ \)

\(det⁡ A = ad-bc \)

Example:

Find determinant of 2x2 matrix A

 

\(det A = \begin{vmatrix} 4 & 12 \\ 2 & 7 \end{vmatrix} \\ \)

 

Solution:

 

\(|A| = (7)(4) – (2)(12)\)

 

\(|A| = 28 – 24\)

 

\(|A| = 4\)

 

For 3x3 Matrix Multiplications:

For the calculations of matrix A = (aij)3×3 from expansion of column is determined by the following formula:

 

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

 

\(det⁡ A= a\begin{vmatrix} e & f \\h & i\end{vmatrix}  - d\begin{vmatrix}b & c \\h & i\end{vmatrix}+g\begin{vmatrix}b & c \\e & f\end{vmatrix} \)

 

Example:

 

\(det A = \begin{vmatrix} 2 & 0 & 3\\1 & 4 & 1 \\0 & 4 & 7 \end{vmatrix} \\ \)

 

Solution:

 

\(det⁡ A= 2\begin{vmatrix} 4 & 1 \\4 & 7\end{vmatrix}  - 1\begin{vmatrix}0 & 3 \\4 & 7\end{vmatrix}+0\begin{vmatrix}0 & 3 \\4 & 1\end{vmatrix} \)

 

\( det⁡ A = 2[(7)(4)-(4)(1)]-1[(4)(3)-(7)(0)]+ 0[(4)(3)-(1)(0)] \)

 

\( det⁡ A = 2[28-4]-1[12-0]+ 0[12-0] \)

 

\( det⁡ A = 2[24]-1[12]+ 0[12] \)

 

\( det⁡ A = 48-12+ 0 \)

 

\( det⁡ A = 36 \)

For 4x4 Matrix Multiplications:

For the calculations of matrix A = (aij)4×4 from expansion of column is determined by the following formula:

 

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

 

\(det⁡ A= a\begin{vmatrix} f & g  & h\\j & k & l\\n & o & p\end{vmatrix}  - e\begin{vmatrix}b & c & d\\j & k & l\\ n & o & p\end{vmatrix}+i\begin{vmatrix}b & c & d \\f & g & h\\n & o & p\end{vmatrix}-m\begin{vmatrix}b & c & d\\f & g & h\\j & k & l\end {vmatrix}\)

 

Then, simply determine the determinant of 3x3 by using above formula of 3x3.

Example:

\(det A = \begin{vmatrix} 1 & 8 & 7 & 2\\2 & 4 & 3 &8 \\1 & 4 & 3 & 2 \\ 1 & 4 & 9 & 6 \end{vmatrix} \\ \)

Solution:

\(det⁡ A= 1\begin{vmatrix}4 & 3  & 8\\4 & 3 & 2\\4 & 9 & 6\end{vmatrix}  - 2\begin{vmatrix}8 & 7 & 2\\4 & 3 & 2\\ 4 & 9 & 6\end{vmatrix}+1\begin{vmatrix}8 & 7 & 2 \\4 & 3 & 8\\4 & 9 & 6\end{vmatrix}-1\begin{vmatrix}8 & 7 & 2\\4 & 3 & 8\\4 & 3 & 2\end {vmatrix}\)

 

\(det⁡ A=1( 4\begin{vmatrix} 3 & 2 \\9 & 6\end{vmatrix}  - 3\begin{vmatrix}4 & 2 \\4 & 6\end{vmatrix}+8\begin{vmatrix}4 & 3 \\4 & 9\end{vmatrix}) -2( 8\begin{vmatrix} 3 & 2 \\9 & 6\end{vmatrix}  - 7\begin{vmatrix}4 & 2 \\4 & 6\end{vmatrix}+2\begin{vmatrix}4 & 3 \\4 & 9\end{vmatrix}) +1( 8\begin{vmatrix}3 & 8 \\9 & 6\end{vmatrix}  - 7\begin{vmatrix}4 & 8 \\4 & 6\end{vmatrix}+2\begin{vmatrix}4 & 3 \\4 & 9\end{vmatrix}) -1( 8\begin{vmatrix} 3 & 8 \\3 & 2\end{vmatrix}  - 7\begin{vmatrix}4 & 8 \\4 & 6\end{vmatrix}+2\begin{vmatrix}4 & 3 \\4 & 3\end{vmatrix})\)

 

\(det⁡ A = 1[4(18-18)-3(24-8)+ 8(36-12)]-2[ 8(18-18)-7(24-8)+ 2(36-12)]+ 1[ 8(18-72)-7(24-32)+2(36-12)] -1[8(6-24)-7(8-32)+ 2(12-12)]\)

 

\(det⁡ A = 1[4(0)-3(16)+ 8(24)]-2[ 8(0)-7(16)+ 2(24)]+ 1[ 8(-54)-7(-8)+ 2(24)]-1[8(-18)-7(-24)+ 2(0)]\)

 

\(det⁡ A = 1[0-48+192]-2[0-112+48]+ 1[ -432+56+48]-1[-144+168+0]\)

 

\(det⁡ A = 1[144]-2[-64]+ 1[-328]-1[24]\)

 

\(det⁡ A = 144+128-328- 24\)

 

\(det⁡ A = -80\)

For 5x5 Matrix Multiplications:

For the calculations of matrix A = (aij)5×5 from expansion of column is determined by the following formula:

\( det A = \begin{vmatrix} a & b & c & d & e\\f & g & h & i & j\\k & l & m & n & o \\ p & q & r & s & t \\ u & v & w & x & y \end{vmatrix} \\ \)

\(det⁡ A= a\begin{vmatrix} g & h & i & j\\l & m & n & o\\q & r & s & t\\v & w & x & y\end{vmatrix} - f\begin{vmatrix}b & c & d & e\\l & m & n & o\\ q & r & s & t\\ v & w & x & y\end{vmatrix}+k\begin{vmatrix}b & c & d & e \\ g & h & i & j\\q & r & s & t\\v & w & x & y\end{vmatrix}-p\begin{vmatrix}b & c & d & e\\g & h & i & j\\l & m & n & o\\q & r & s & t\end {vmatrix}\\ \)