T20 - Week2 Discussion

T20 Suppose that A is a square matrix of size n and alpha is a scalar. Prove that det (alpha * A) = alpha^n det (A).

matrix_elements = c(3,2,1,5,1,0,2,1,2,3,7,2,8,4,4,0)
MatrixA = matrix(matrix_elements,nrow=4, ncol=4)
alpha = 3  
print(MatrixA)
##      [,1] [,2] [,3] [,4]
## [1,]    3    1    2    8
## [2,]    2    0    3    4
## [3,]    1    2    7    4
## [4,]    5    1    2    0
detofA = det(MatrixA)
detofalphaXA = det(MatrixA * 3)
print(detofalphaXA)
## [1] 21384
result = detofA * `^`(alpha,4)
print(result)
## [1] 21384

This proves that the det (alpha * A) = alpha^n det (A)