Find the determinant

Question

1+x   1       1     1

1      1-x     1     1

1       1    1+z    1

1       1      1      1-z

Answer 1

1+x    1    1     1               x    0     -z      0  added -row 3 to row 1

  1    1-x   1     1     --->    0   -x    -z      0  added -row 3 to row 2

  1      1   1+z  1               1     1    1+z   1

  1      1    1    1-z             z     z      z²    0   added -(1-z)row 3 to                                                                                         row 4

 

add (1/z) row 4 to row 1           x+1     1     0      0

add (1/z) row 4 to row 2 --->      1     1-x    0      0

                                                  1       1   1+z    1

                                                  z       z      z²    0

 

This is a "block matrix" whose determinant is the same as the determinant of the given matrix.

 

Since there is a 2 x 2 zero matrix in the upper right corner, The determinant of the block matrix is (detA)(detC), where A is the

2 x 2 matrix in the upper left corner and C is the 2 x 2 matrix in the lower right corner.  

 

det A = (x+1)(1-x) - 1 = -x² and det C = (1+z)(0)-(z²)(1) = -z²

 

So, the determinant of the given matrix = (-x²)(-z²) = x²z²

 

 

 

 

Answer 2

https://www.khanacademy.org/math/linear-algebra/matrix_transformations/determinant_depth/v/linear-algebra-simpler-4x4-determinant