Matrices help solve simultaneous equations and are most often found in problems related to electronics, robotics, statics, optimization, linear programming and genetics. It is best to use computers to solve a large system of equations. However, you can solve for the determinant of a 4-by-4 matrix by replacing the values in the rows and using the "upper triangular" form of matrices. This states that the determinant of the matrix is the product of the numbers in the diagonal when everything below the diagonal is a 0.

Write down the rows and columns of the 4-by-4 matrix -- between to vertical lines -- to find the determinant. For example:

Row 1 |1 2 2 1|

Row 2 |2 7 5 2|

Row 3 |1 2 4 2|

Row 4 |-1 4 -6 3|

Replace the second row to create a 0 in the first position, if possible. The rule states that (row j) + or - (C * row i) will not change the determinant of the matrix, where "row j" is any row in the matrix, "C" is a common factor and "row i" is any other row in the matrix. For the example matrix, (row 2) - (2 * row 1) will create a 0 in the first position of row 2. Subtract the values of row 2, multiplied by each number in row 1, from each corresponding number in row 2. The matrix becomes:

Row 1 |1 2 2 1|

Row 2 |0 3 1 0|

Row 3 |1 2 4 2|

Row 4 |-1 4 -6 3|

Replace the numbers in the third row to create a 0 in both the first and second positions, if possible. Use a common factor of 1 for the example matrix, and subtract the values from the third row. The example matrix becomes:

Row 1 |1 2 2 1|

Row 2 |0 3 1 0|

Row 3 |0 0 2 1|

Row 4 |-1 4 -6 3|

Replace the numbers in the fourth row to get zeroes in the first three positions, if possible. In the example problem the last row has -1 in the first position and the first row has a 1 in the corresponding position, so add the multiplied values of the first row to the corresponding values of the last row to get a zero in the first position. The matrix becomes:

Row 1 |1 2 2 1|

Row 2 |0 3 1 0|

Row 3 |0 0 2 1|

Row 4 |0 6 -4 4|

Replace the numbers in the fourth row again to get zeroes in the remaining positions. For the example, multiply the second row by 2 and subtract the values from those of the last row to convert the matrix to an "upper triangular" form, with only zeroes below the diagonal. The matrix now reads:

Row 1 |1 2 2 1|

Row 2 |0 3 1 0|

Row 3 |0 0 2 1|

Row 4 |0 0 -6 4|

Replace the numbers in the fourth row again to get zeroes in the remaining positions. Multiply the values in the third row by 3, then add them to the corresponding values in the last row to get the final zero below the diagonal in the example matrix. The matrix now reads:

Row 1 |1 2 2 1|

Row 2 |0 3 1 0|

Row 3 |0 0 2 1|

Row 4 |0 0 0 7|

Multiply the numbers in the diagonal to solve for the determinant of the 4-by-4 matrix. In this case, multiply 1*3*2*7 to find a determinant of 42.