TheWrathOfMath
Junior Member
- Joined
- Mar 31, 2022
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Prove or disprove using a counterexample:
If A is a square matrix and det(A)=/=0, then det(adj(A))=/=0.
My work:
I know that if A is an invertible matrix (detA=/=0), then adj(A) is also invertible since [math]A*adj(A)=det(A)I[/math] => adj(A) is invertible since det(A)=/=0. This means that det(adj(A))=/=0.
Since I am given that A is a square matrix whose determinant is nonzero, I know that A is invertible, hence the above statement is correct, which means that the original claim I am trying to prove is correct.
Is what I did alright?
If A is a square matrix and det(A)=/=0, then det(adj(A))=/=0.
My work:
I know that if A is an invertible matrix (detA=/=0), then adj(A) is also invertible since [math]A*adj(A)=det(A)I[/math] => adj(A) is invertible since det(A)=/=0. This means that det(adj(A))=/=0.
Since I am given that A is a square matrix whose determinant is nonzero, I know that A is invertible, hence the above statement is correct, which means that the original claim I am trying to prove is correct.
Is what I did alright?
Last edited: