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(D) Let us analyze each statement:$(i)$ If $A$ is a symmetric matrix,then $A^T = A$. The adjoint of $A$ is defined as the transpose of the cofactor ma

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None of these

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(D) Let us analyze each statement:$(i)$ If $A$ is a symmetric matrix,then $A^T = A$. The adjoint of $A$ is defined as the transpose of the cofactor matrix. It can be shown that $adj(A^T) = (adj\,A)^T$. Since $A^T = A$,we have $adj(A) = (adj\,A)^T$,which means $adj(A)$ is symmetric. This statement is true.$(ii)$ If $A = I$ (unit matrix),then $adj(I) = |I|I^{-1} = 1 	imes I = I$. Thus,the adjoint of a unit matrix is a unit matrix. This statement is true.$(iii)$ By the fundamental property of the adjoint of a matrix,$A(adj\,A) = (adj\,A)A = |A|I$. This statement is true.$(iv)$ If $A$ is a diagonal matrix,its cofactors will result in another diagonal matrix. Thus,the adjoint of a diagonal matrix is a diagonal matrix. This statement is true.Since all statements $(i), (ii), (iii),$ and $(iv)$ are correct,none of the statements are incorrect. Therefore,the correct option is $(d)$.

Which of the following statements is/are incorrect?
$(i)$ Adjoint of a symmetric matrix is symmetric.
$(ii)$ Adjoint of a unit matrix is a unit matrix.
$(iii)$ $A(adj\,A) = (adj\,A)A = |A|I$.
$(iv)$ Adjoint of a diagonal matrix is a diagonal matrix.

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Which of the following statements is/are incorrect?$(i)$ Adjoint of a symmetric matrix is symmetric.$(ii)$ Adjoint of a unit matrix is a unit matrix.$(iii)$ $A(adj\,A) = (adj\,A)A = |A|I$.$(iv)$ Adjoint of a diagonal matrix is a diagonal matrix.