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OscarGiCast

pythonic_human 🐍

Row Operations on Determinants

/ 4 min read

linear systems, matrix, determinant, numpy
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The following row operations affects the determinant of a matrix but NOT THE MATRIX SINGULARITY. We are going to make operations on the matrix MM with det(M)=8det(M) = 8:

M=[3451123252355236]M = \begin{bmatrix} 3 & 4 & 5 & 1 \\ 1 & 2 & 3 & 2 \\ 5 & 2 & 3 & 5 \\ 5 & 2 & 3 & 6 \end{bmatrix} 
M = np.array([
    [3, 4, 5, 1],
    [1, 2, 3, 2],
    [5, 2, 3, 5],
    [5, 2, 3, 6],
], dtype="float32")
print(M)
det_M = np.linalg.det(M)
print(f"{det_M=}"

Output:

Terminal window
[[3. 4. 5. 1.]
 [1. 2. 3. 2.]
 [5. 2. 3. 5.]
 [5. 2. 3. 6.]]
det_M=8.0

Swapping 2 rows

Swapping one position 2 rows of a matrix, changes the sign of its determinant.

r1↔r2r_{1} \leftrightarrow r_{2} M=[3451123252355236]→r1↔r2M1=[3451523512325236]M = \begin{bmatrix} 3 & 4 & 5 & 1 \\ \textcolor{green}{1} & \textcolor{green}{2} & \textcolor{green}{3} & \textcolor{green}{2} \\ \textcolor{red}{5} & \textcolor{red}{2} & \textcolor{red}{3} & \textcolor{red}{5} \\ 5 & 2 & 3 & 6 \end{bmatrix} \xrightarrow{r_{1} \leftrightarrow r_{2}} M_{1} = \begin{bmatrix} 3 & 4 & 5 & 1 \\ \textcolor{red}{5} & \textcolor{red}{2} & \textcolor{red}{3} & \textcolor{red}{5} \\ \textcolor{green}{1} & \textcolor{green}{2} & \textcolor{green}{3} & \textcolor{green}{2} \\ 5 & 2 & 3 & 6 \end{bmatrix} det(M1)=−det(M)=−8det(M_{1}) = -det(M) = -8 
# Row Swapping: 1 position
# Swapping row 1 with row 2.
# |M| * -1
M1 = M.copy()
M1[[2, 1]] = M1[[1, 2]]
print(M1)
det_M1 = np.linalg.det(M1)
print(f"{det_M1=} is -1 * {det_M=}"

Output:

Terminal window
[[3. 4. 5. 1.]
 [5. 2. 3. 5.]
 [1. 2. 3. 2.]
 [5. 2. 3. 6.]]
det_M1=-8.0 is -1 * det_M=8.0

Swapping two positions 4 rows of a matrix, changes the sign of its determinant.

r0↔r1r2↔r3r_{0} \leftrightarrow r_{1} \\ r_{2} \leftrightarrow r_{3} M=[3451123252355236]→r0↔r1M1=[1232345152355236]M = \begin{bmatrix} \textcolor{red}{3} & \textcolor{red}{4} & \textcolor{red}{5} & \textcolor{red}{1} \\ \textcolor{green}{1} & \textcolor{green}{2} & \textcolor{green}{3} & \textcolor{green}{2} \\ 5 & 2 & 3 & 5 \\ 5 & 2 & 3 & 6 \end{bmatrix} \xrightarrow{r_{0} \leftrightarrow r_{1}} M_{1} = \begin{bmatrix} \textcolor{green}{1} & \textcolor{green}{2} & \textcolor{green}{3} & \textcolor{green}{2} \\ \textcolor{red}{3} & \textcolor{red}{4} & \textcolor{red}{5} & \textcolor{red}{1} \\ 5 & 2 & 3 & 5 \\ 5 & 2 & 3 & 6 \end{bmatrix} det(M1)=det(M)∗−1det(M_{1}) = det(M) * -1 M1=[1232345152355236]→r2↔r3M2=[1232345152365235]M_{1} = \begin{bmatrix} 1 & 2 & 3 & 2 \\ 3 & 4 & 5 & 1 \\ \textcolor{green}{5} & \textcolor{green}{2} & \textcolor{green}{3} & \textcolor{green}{5} \\ \textcolor{red}{5} & \textcolor{red}{2} & \textcolor{red}{3} & \textcolor{red}{6} \end{bmatrix} \xrightarrow{r_{2} \leftrightarrow r_{3}} M_{2} = \begin{bmatrix} 1 & 2 & 3 & 2 \\ 3 & 4 & 5 & 1 \\ \textcolor{red}{5} & \textcolor{red}{2} & \textcolor{red}{3} & \textcolor{red}{6} \\ \textcolor{green}{5} & \textcolor{green}{2} & \textcolor{green}{3} & \textcolor{green}{5} \end{bmatrix} det(M2)=det(M1)∗−1=(det(M)∗−1)∗−1=det(M)=8det(M_{2}) = det(M_{1}) * -1 = (det(M) * -1) * -1 = det(M) = 8 
# Row Swapping: 2 positions
# Swapping row 0 with row 1.
# Swapping row 2 with row 3.
# |M| * -1 * -1
M1 = M.copy()
M1[[1, 0]] = M1[[0, 1]]
M1[[3, 2]] = M1[[2, 3]]
print(f"{np.linalg.det(M1)}"

Output:

Terminal window
8.0

Row Scaling

Scaling one row of a matrix by a scalar, scales the determinant of the matrix by the same scalar.

r0→k∗r0det(A)=k∗det(M)r_{0} \rightarrow k*r_{0} \\ det(A) = k * det(M) M=[3451123252355236]→A=[3k4k5kk123252355236]M = \begin{bmatrix} \textcolor{red}{3} & \textcolor{red}{4} & \textcolor{red}{5} & \textcolor{red}{1} \\ 1 & 2 & 3 & 2 \\ 5 & 2 & 3 & 5 \\ 5 & 2 & 3 & 6 \end{bmatrix} \xrightarrow{} A = \begin{bmatrix} \textcolor{red}{3k} & \textcolor{red}{4k} & \textcolor{red}{5k} & \textcolor{red}{k} \\ 1 & 2 & 3 & 2 \\ 5 & 2 & 3 & 5 \\ 5 & 2 & 3 & 6 \end{bmatrix} det(A)=8kdet(A) = 8k 
# Row scaling: by k
k = 3
A = M.copy()
A[0] = k * A[0]
print(A)
det_A = np.linalg.det(A)
print(f"{det_A = } is {k} times {det_M = }")

Output:

Terminal window
[[ 9. 12. 15.  3.]
 [ 1.  2.  3.  2.]
 [ 5.  2.  3.  5.]
 [ 5.  2.  3.  6.]]
det_A = 24.0 is 3 times det_M = 8.0

Row Addition

The adition of a row with another row scaled by a scalar, does not affect the determinant of the matrix.

r0→k1r1+k2r2+...r0→∑i=1nkirir_{0} \rightarrow k_{1}r_{1} + k_{2}r_{2} + ... \\ r_0 \rightarrow \sum_{i=1}^{n} k_i r_i

Where:

  • r0r_0 is the original row.
  • rir_i are the rows to be added to r0r_0 with coefficients kik_i
  • kik_i are the coefficients of the rows rir_i, where i=1,2,...,ni = 1, 2, ..., n
  • nn is the total number of rows.

And the equation for the determinant remains the same:

det(A)=det(M)\text{det}(A) = \text{det}(M) 
# Row addition:
A = M.copy()
A[0] = A[0] + 2*A[1] + 3*A[2]
print(A)
det_A = np.linalg.det(A)
print(f"{det_A = } is the same as {det_M = }")

Output:

Terminal window
[[20. 14. 20. 20.]
 [ 1.  2.  3.  2.]
 [ 5.  2.  3.  5.]
 [ 5.  2.  3.  6.]]
det_A = 8.0 is the same as det_M = 8.0