Choose the set of correct options. If A is an upper triangular 3×3 matrix, then the adjoint matrix of A is also an upper triangular matrix. If A is an invertible upper triangular 3... Choose the set of correct options. If A is an upper triangular 3×3 matrix, then the adjoint matrix of A is also an upper triangular matrix. If A is an invertible upper triangular 3×3 matrix, then the inverse matrix of A is also an upper triangular matrix. Let A be an arbitrary real 3×3 matrix. If C is the adjoint matrix of A, then C is also the adjoint matrix of A^T. Cjk denotes the cofactor with respect to the j-th row and the k-th column of a 3×3 matrix A. If another matrix B is obtained from A by replacing the j-th row of A with [3, 0, 0], then det(B) = 3Cjk. If A is an invertible 3×3 matrix and C=adj(adj(A)), then det(C)=det(A)^9.
Understand the Problem
The question is asking to choose the correct statements regarding properties of upper triangular matrices and their adjoint and inverse matrices. It involves understanding linear algebra concepts specifically relating to matrices.
Answer
All options are correct.
All options are correct.
Answer for screen readers
All options are correct.
More Information
All given statements are supported by known properties of triangular matrices, determinants, and adjugate matrices. An upper triangular matrix's inverse and adjugate retain triangularity. If operations on a matrix affect only a row or column, specific resultant determinant consequences follow. The statement about adj(adj(A)) hinges on the determinant properties.
Tips
A common mistake might be assuming results for general matrices apply to triangular ones without accounting for their specific properties.
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