Prove help(Linear Algebra)?

Question

Prove that if A is an n x n matrix and nonsingular matrix

then, det A^-1 = 1/(det A)

Answer 1

To prove that det(A)*det(B) = det(AB) Iwould suggest to use the 3 basic operations that can be done by matrix multiplication:
1. Switch two rows(or columns) - negates the determinants sign.
2. Multiplying a row(or a column) by a scalar - multiply the determinant by that scalar.;
3. Adding a row multiplied by a scalar to another row - no change.

Every square matrix is a product of those elementart matrix, and the product of their determinants is the determinant of their product.

1=det(I) = det(A * A^-1) = det(A)*det(A^-1)
Multiply by det(A)^-1 and you'll get

det(A)^-1 = det(A^-1)

Answer 2

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Answer 3

Can you use the property that:

det(AB) = (detA)(detB) ?

if so, then you got it since the det of the identity is 1

+add
Carla misunderstood the question. You asked about the deteminant of a matrix, not the inverse of a real number.

Answer 4

Look, any number with a negative exponent can be turned into 1/number.
In this case, your exponent is -1
So, you turn the number around and you get 1/(det A)^1

This is actually a mathematical rule, so....