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why is det(kA)=k^ndet(A)?
[show that it is true for all n and all k]
[show that it is true for all n and all k]
The determinant of kA is the product of all the eigenvalues of the matrix kA. In other words, it is a scalar value that represents the scaling factor of the matrix kA.
The relationship between det(kA) and det(A) is that det(kA) is equal to k raised to the power of n, where n is the dimension of the matrix, multiplied by det(A). In other words, the determinant of a scaled matrix is equal to the scaling factor raised to the power of the matrix's dimension, multiplied by the determinant of the original matrix.
Yes, the proof of det(kA)=k^n det(A) is applicable to all matrices, as long as the matrix is square and the scaling factor k is a constant.
Yes, this proof can be extended to matrices with complex numbers, as long as the complex numbers are treated as scalars and follow the same rules of multiplication and addition as real numbers.
This proof can be used in various practical applications, such as in solving systems of linear equations, calculating the area of a parallelogram or the volume of a parallelepiped, and in determining the invertibility of a matrix. It is also used in various fields of science and engineering, such as quantum mechanics, circuit analysis, and computer graphics.