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A square matrix A is called a diagonal matrix if all its off-diagonal elements are equal to zero, i.e., a_{ij} = 0 for all i ≠ j. A square matrix A is called an upper triangular matrix if all the elements below the main diagonal are equal to zero, i.e., a_{ij} = 0 for all i > j. Similarly, A is called a lower triangular matrix if all the elements above the main diagonal are equal to zero, i.e., a_{ij} = 0 for all i < j.

The adjugate of a square matrix A, denoted by adj(A), is a matrix whose elements are the signed minors of A, where the minors are found by deleting a row and a column and finding the determinant of the remaining matrix. The (-1)^(i+j) coefficient sign determines the sign of the corresponding minor.

Let us consider an n x n diagonal matrix A. Since all off-diagonal elements are zero, the adjugate of A can be found as follows: 1. For a diagonal element a_{ii}, the corresponding minor is obtained by deleting the i-th row and i-th column, resulting in an (n-1) x (n-1) diagonal matrix. The determinant of this minor will be the product of its diagonal elements. 2. For an off-diagonal element a_{ij} with i ≠ j, the corresponding minor is obtained by deleting the i-th row and j-th column, resulting in an (n-1) x (n-1) matrix with one less non-zero diagonal element. The determinant of this minor will always be zero. Thus, all off-diagonal elements of adj(A) are zero, and A remains a diagonal matrix.

Let us consider an n x n upper triangular matrix A. Since all elements below the main diagonal are zero, the adjugate of A can be found as follows: 1. For a diagonal element a_{ii}, the corresponding minor is obtained by deleting the i-th row and i-th column, resulting in an (n-1) x (n-1) upper triangular matrix. The determinant of this minor will be the product of its diagonal elements. 2. For an off-diagonal element above the main diagonal a_{ij} with i < j, the corresponding minor is obtained by deleting the i-th row and j-th column, resulting in an (n-1) x (n-1) matrix with a zero row - row i. Since the determinant of a matrix with a zero row is zero, the determinant of this minor will always be zero. 3. For an off-diagonal element below the main diagonal a_{ij} with i > j, the corresponding minor is obtained by deleting the i-th row and j-th column, resulting in an (n-1) x (n-1) upper triangular matrix. The determinant of this minor will be the product of its diagonal elements. Thus, all off-diagonal elements of adj(A) above the main diagonal are zero, and A remains an upper triangular matrix. A similar argument can be made for a lower triangular matrix, showing that adj(A) remains lower triangular in that case as well. In conclusion, if A is a diagonal or triangular matrix, then adj(A) is also diagonal or triangular, respectively.