+0

# Determinant of a matrix

0
364
2

Hi. I need help at determinating the determinant of this matrix.

3  -5  -2  2

-4   7   4  4

4  -9  -3  7

2  -6  -3  2

I calculated and it gave me 0 but the result should be 27.

Oct 16, 2017

#1
+95866
+1

3  -5  -2  2

-4   7   4  4

4  -9  -3  7

2  -6  -3  2

We can use some row operations to minimize the work needed.....

Add the 2nd row to the 3rd row.....put the result  in the 3rd row

3  -5  -2  2

-4   7   4  4

0  -2   1 11

2  -6  -3  2

Multiply the 4th row by 2.......add it to the second row

3  -5  -2  2

0  -5  -2  8

0  -2   1 11

2  -6  -3  2

Expand along the first row and first column

3 [  -5  -2   8

-2   1  11

-6  -3  2 ]

Rewrite the first two columns of the matrix

3 [   -5   -2   8    -5     -2

-2    1   11   -2     1

- 6    -3   2    -6    -3 ]

3  [  determinant of the marix]  =

3  [  (-5* 1 *2  +  -2 * 11 * -6  +  8 * -2 *-3) - ( -6*1*8 + -3*11*-5 + 2 *-2 * -2) ] =  135   (1)

-2  [-5  -2  2

-5  -2  8

-2   1 11

Rewrite the first two columns

-2 [  -5  -2    2     -5   -2

-5  -2   8     -5    -2

-2   1   11    -2    1 ]

-2 [ determinant of the matrix ]

-2 [ ( -5*-2*11 + -2*8*-2 + 2*-5*1) - (-2*-2*2 + 1*8*-5 + 11*-5*-2) ]  =  -108   (2)

Add (1)  and (2)  =    135  - 108   =    27

Oct 16, 2017
edited by CPhill  Oct 17, 2017
#2
+21191
+1

Determinant of a matrix

3  -5  -2  2

-4   7   4  4

4  -9  -3  7

2  -6  -3  2

1.  tridiagonal matrix, e. g. Gauß
$$\begin{pmatrix} 3 & -5 & -2 & 2 \\ 0 & \frac{1}{3} & 1\ \frac{1}{3} & 6\ \frac{2}{3} \\ 0 & 0 & 9 & 51 \\ 0 & 0 & 0 & 3 \\ \end{pmatrix}$$

2.  Determinant: multiply the diagonally elements

The determinant of this matrix is $$3 \cdot \frac{1}{3}\cdot 9\cdot 3 = \mathbf{27 }$$

Oct 17, 2017