Determinant of 4×4 Matrix

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Determinant of 4×4 Matrix

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Determinant of a 4×4 matrix is a unique number which is calculated using a particular formula. If a matrix order is n x n, then it is a square matrix. Hence, here 4×4 is a square matrix which has four rows and four columns. If A is square matrix then the determinant of matrix A is represented as |A|.

To find the determinant of a 4×4 matrix, we will use the simple method, which we usually use to find the determinant of a 3×3 matrix.

How to calculate determinant of 4×4 matrix?

Before we calculate the determinant of a matrix of order 4, let us first check a few conditions.

  • if there is any condition, where determinant could be 0 (for example, the complete row or complete column is 0)
  • if factoring out of any row or column is possible.
  • If the elements of the matrix are the same but reordered on any column or row.

If any of the three cases given above is met, the corresponding methods for calculating 3×3 determinants are used. We transform a row or a column to fill it with 0, except for one element. The determinant will be equivalent to the product of that element and its cofactor. In this situation, the cofactor is a 3×3 determinant, which is estimated with its particular formula.

For Example:

\(\left|\begin{array}{cccc}1 & 4 & 2 & 3 \\ 0 & 1 & 4 & 4 \\ -1 & 0 & 1 & 0 \\ 2 & 0 & 4 & 1\end{array}\right|=1\left|\begin{array}{ccc}1 & 4 & 4 \\ 0 & 1 & 0 \\ 0 & 4 & 1\end{array}\right|-4\left|\begin{array}{ccc}0 & 4 & 4 \\ -1 & 1 & 0 \\ 2 & 4 & 1\end{array}\right|+2\left|\begin{array}{ccc}0 & 1 & 4 \\ -1 & 0 & 0 \\ 2 & 0 & 1\end{array}\right|-3\left|\begin{array}{ccc}0 & 1 & 4 \\ -1 & 0 & 1 \\ 2 & 0 & 4\end{array}\right|\)

Also, read:

Solved Examples

Let us solve some examples here.

Example 1:

\(\left|\begin{array}{llll}1 & 2 & 6 & 6 \\ 4 & 7 & 3 & 2 \\ 0 & 0 & 0 & 0 \\ 1 & 2 & 2& 9\end{array}\right|\)

As we can see in the above example, the elements in third row is all 0. Hence, the value of determinant will be zero.

Example 2:

\(\left|\begin{array}{llll}2 & 1 & 2 & 3 \\ 6 & 7 & 6 & 9 \\ 0 & 6 & 0 & 0 \\ 1 & 2 & 1 & 4\end{array}\right|\)

As we can see here, column C1 and C3 are equal. Therefore, the determinant of the matrix is 0.

Example 3:

\(\left|\begin{array}{cccc}1 & 2 & 3 & 4 \\ 2 & 5 & 7 & 3 \\ 4 & 10 & 14 & 6 \\ 3 & 4 & 2 & 7\end{array}\right|\)

As we can see here, second and third rows are proportional to each other. Hence, the determinant of the matrix is 0.

Example 4:

|A| = \(\left|\begin{array}{cccc}4 & 3 & 2 & 2 \\ 0 & 1 & -3 & 3 \\ 0 & -1 & 3 & 3 \\ 0 & 3 & 1 & 1\end{array}\right|\)

As we can see, there is only one element other than 0 on first column, therefore we will use the general formula using this column. The cofactors of the elements which are 0 are not required to be evaluated because the product of cofactors and the elements will be 0 here.

|A| =4(1×3×1+(−1)×1×3+3×(−3)×3−(3×3×3+3×1×1+1×(−3)×(−1)))

=4(3-3-27-(27+3+3))

=4×(-60)

= -240