Question

Which of the following is/are incorrect
(i) Adjoint of a symmetric matrix is symmetric,
(ii) Adjoint of a unit matrix is a unit matrix,
(iii) A(adjA)=(adjA)A=|A|I and
(iv) Adjoint of a diagonal matrix is a diagonal matrix
A. (i)
B. (ii)
C. (iii) and (iv)
D. None of these

Answer 1

Hint: Check whether the option satisfies the properties of the adjoint of a matrix. By taking the transpose of the cofactors matrix, a matrix's adjoint is created. The adjoint of a matrix's properties can be used for any square matrix of order $n\times n$.

Formula Used:
Property of Adjoint Matrix:
$(adjA)^T=adj(A^T)$

Complete step-by-step solution:
Let’s check option (i) we have, Adjoint of a symmetric matrix is symmetric.
Suppose $A$ is a symmetric matrix.
Then, $(A^T)=A$
According to the property of adjoint of a matrix;
$(adjA)^T=adj(A^T) \\
(adjA) ^T=adjA \:[\because (A^T)=A]$
Hence, the given statement is correct Adjoint of a symmetric matrix is also symmetric.

Let’s check option (ii) we have, Adjoint of a unit matrix is a unit matrix.
Suppose $A$ be a unit matrix.
Then,$ (A^T)=A$
According to the property of adjoint of a matrix;
$(adjA^T)=adj(A)^T \\
adjA =(adjA) ^T\:[\because (A^T)=A]$
Hence, the given statement is correct Adjoint of a unit matrix is a unit matrix.

Let’s check option (iii) we have, $A(adjA)=(adjA)A=|A|I$
Suppose $A$ be a non-singular matrix.
We know that the inverse of a non-singular matrix is given by;
$A^{-1}=\dfrac{adjA}{|A|}$
Multiplying both sides with $A$ we get;
$A.A^{-1}=\dfrac{A(adjA)}{|A|}\\
\Rightarrow A(adjA)=|A|I\\
\Rightarrow (adjA)A=|A|I$
Hence, the given statement is correct A(adjA)=(adjA)A=|A|I.

Let’s check option (iv) we have, Adjoint of a diagonal matrix is a diagonal matrix.
We know that this is the property of a diagonal matrix that the adjoint of a diagonal matrix is also a diagonal matrix.
All of the above-mentioned statements are correct.

So, option D is correct.

Note: The adjoint of a matrix is formed by taking the transpose of the cofactors' matrix.
$adj(A^T) = adj(A)^T$, where $A^T$ is a matrix transposed from $A$.
If you multiply a matrix $A$ by its adjoint, you get a diagonal matrix whose diagonal entries are the determinant $det (A)$. $I$ is an identity matrix, and $A adj(A) = adj(A) A = det(A) I$.