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Category : JEE Main & Advanced

(1) Symmetric matrix : A square matrix \[A=[{{a}_{ij}}]\]is called symmetric matrix if \[{{a}_{ij}}={{a}_{ji}}\]for all i, j or \[{{A}^{T}}=A\].

Example : \[\left[ \begin{matrix} a & h & g  \\ h & b & f  \\ g & f & c  \\ \end{matrix} \right]\]

(2) Skew-symmetric matrix : A square matrix \[A=[{{a}_{ij}}]\]is called skew- symmetric matrix if \[{{a}_{ij}}=-{{a}_{ji}}\]for all i, j or \[{{A}^{T}}=-A\].

Example : \[\left[ \begin{matrix} 0 & h & g  \\ -h & 0 & f  \\ -g & -f & 0  \\ \end{matrix} \right]\]

All principal diagonal elements of a skew- symmetric matrix are always zero because for any diagonal element.

\[{{a}_{ij}}=-{{a}_{ij}}\Rightarrow {{a}_{ij}}=0\]

Properties of symmetric and skew-symmetric matrices

(i) If A is a square matrix, then \[A+{{A}^{T}},A{{A}^{T}},{{A}^{T}}A\] are symmetric matrices, while \[A-{{A}^{T}}\]is skew- symmetric matrix.

(ii) If A is a symmetric matrix, then\[-A,KA,{{A}^{T}},{{A}^{n}},{{A}^{-1}},{{B}^{T}}AB\] are also symmetric matrices, where \[n\in N\], \[K\in R\] and B is a square matrix of order that of A.

(iii) If A is a skew-symmetric matrix, then

(a) \[{{A}^{2n}}\]is a symmetric matrix for \[n\in N\].

(b) \[{{A}^{2n+1}}\]is a skew-symmetric matrix for \[n\in N\].

(c) kA is also skew-symmetric matrix, where \[k\in R\].

(d)  \[{{B}^{T}}AB\] is also skew- symmetric matrix where B is a square matrix of order that of A.

(iv) If A, B are two symmetric matrices, then

(a)  \[A\pm B,\,\,AB+BA\] are also symmetric matrices,

(b)  \[AB-BA\]is a skew- symmetric matrix,

(c)   AB is a symmetric matrix, when \[AB=BA\].

(v) If A, B  are two skew-symmetric matrices, then

(a) \[A\pm B,\,\,AB-BA\] are skew-symmetric matrices,

(b) \[AB+BA\]is a symmetric matrix.

(vi) If A a skew-symmetric matrix and C is a column matrix, then \[{{C}^{T}}\]AC is a zero matrix.

(vii) Every square matrix A can unequally be expressed as sum of a symmetric and skew-symmetric matrix

i.e., \[A=\left[ \frac{1}{2}(A+{{A}^{T}}) \right]+\left[ \frac{1}{2}(A-{{A}^{T}}) \right]\].

(3) Singular and Non-singular matrix : Any square matrix A is said to be non-singular if \[|A|\ne 0,\]and a square matrix A is said to be singular if \[|A|\,=0\]. Here \[|A|\](or det(A) or simply det  \[|A|\] means corresponding determinant of square matrix A.

Example : \[A=\left[ \begin{matrix} 2 & 3  \\ 4 & 5  \\ \end{matrix} \right]\] then\[|A|\,=\left| \,\begin{matrix} 2 & 3  \\ 4 & 5  \\\end{matrix}\, \right|=10-12=-2\Rightarrow A\] is a non-singular matrix.

(4) Hermitian and Skew-hermitian matrix : A square matrix \[A=[{{a}_{ij}}]\] is said to be hermitian matrix if

\[{{a}_{ij}}={{\bar{a}}_{ji}}\,;\,\,\forall i,j\,\,i.e.,\,A={{A}^{\theta }}\].

Example : \[\left[ \begin{matrix} a & b+ic  \\ b-ic & d  \\ \end{matrix} \right]\,,\,\,\left[ \begin{matrix} 3 & 3-4i & 5+2i  \\ 3+4i & 5 & -2+i  \\ 5-2i & -2-i & 2  \\ \end{matrix} \right]\]

are Hermitian matrices. If A is a Hermitian matrix then \[{{a}_{ii}}={{\bar{a}}_{ii}}\,\,\Rightarrow \]\[{{a}_{ii}}\] is real \[\forall i,\] thus every diagonal element of a Hermitian matrix must be real.

A square matrix, \[A=\,\,|{{a}_{jj}}|\] is said to be a Skew-Hermitian if \[{{a}_{ij}}=-{{\bar{a}}_{ji}}.\,\forall i,\,j\,i.e.\,{{A}^{\theta }}=-A\]. If A is a skew-Hermitian matrix, then \[{{a}_{ii}}=-{{\bar{a}}_{ii}}\Rightarrow {{a}_{ii}}+{{\bar{a}}_{ii}}=0\] i.e. \[{{a}_{ii}}\]must be purely imaginary or zero.

Example : \[\left[ \begin{matrix} 0 & -2+i  \\ 2-i & 0  \\ \end{matrix} \right],\,\,\left[ \begin{matrix} 3i & -3+2i & -1-i  \\ 3+2i & -2i & -2-4i  \\ 1-i & 2-4i & 0  \\ \end{matrix} \right]\]

are skew-hermitian matrices. 

(5) Orthogonal matrix : A square matrix A is called orthogonal if \[A{{A}^{T}}=I={{A}^{T}}A\]  i.e., if \[{{A}^{-1}}={{A}^{T}}\]

Example : \[A=\left[ \begin{matrix} \cos \alpha  & -\sin \alpha   \\ \sin \alpha  & \cos \alpha   \\ \end{matrix} \right]\]is orthogonal because \[{{A}^{-1}}=\left[ \begin{matrix} \cos \alpha  & \sin \alpha   \\ -\sin \alpha  & \cos \alpha   \\ \end{matrix} \right]={{A}^{T}}\]

In fact every unit matrix is orthogonal. Determinant of orthonogal matrix is – 1 or 1.

(6) Idempotent matrix : A square matrix A is called an idempotent matrix if \[{{A}^{2}}=A\].

Example : \[\left[ \begin{matrix} 1/2 & 1/2  \\ 1/2 & 1/2  \\ \end{matrix} \right]\] is an idempotent matrix, because

\[{{A}^{2}}=\left[ \begin{matrix} 1/4+1/4 & 1/4+1/4  \\ 1/4+1/4 & 1/4+1/4  \\ \end{matrix} \right]=\left[ \begin{matrix} 1/2 & 1/2  \\ 1/2 & 1/2  \\ \end{matrix} \right]=A\].

Also, \[A=\left[ \begin{matrix} 1 & 0  \\ 0 & 0  \\ \end{matrix} \right]\text{   and}\,\,B=\left[ \begin{matrix} 0 & 0  \\ 0 & 1  \\ \end{matrix} \right]\] are idempotent matrices because \[{{A}^{2}}=A\] and \[{{B}^{2}}=B\].

In fact every unit matrix is indempotent.

(7) Involutory matrix : A square matrix A is called an involutory matrix if \[{{A}^{2}}=I\,\,\]or \[{{A}^{-1}}=A\]

Example: \[A=\left[ \begin{matrix} 1 & 0  \\ 0 & 1  \\ \end{matrix} \right]\] is an involutory matrix because \[{{A}^{2}}=\left[ \begin{matrix} 1 & 0  \\ 0 & 1  \\ \end{matrix} \right]=I\]

In fact every unit matrix is involutory.

(8) Nilpotent matrix : A square matrix A is called a nilpotent matrix if there exists a \[p\in N\]such that \[{{A}^{p}}=0\].

Example: \[A=\left[ \begin{matrix} 0 & 0  \\ 1 & 0  \\ \end{matrix} \right]\] is a nilpotent matrix because \[{{A}^{2}}=\left[ \begin{matrix} 0 & 0  \\  0 & 0  \\ \end{matrix} \right]=0\],  (Here P = 2)

Determinant of every nilpotent matrix is 0.

(9) Unitary matrix : A square matrix is said to be unitary, if \[\bar{A}'A=\]I since \[|{\bar{A}}'|\,=\,|A|\] and \[|\bar{A}\,'A|\,=\,|\bar{A}\,'||A|\] therefore if \[{\bar{A}}'\] \[A=I,\] we have \[|\bar{A}\,'||A|=1\].

Thus the determinant of unitary matrix is of unit modulus. For a matrix to be unitary it must be non-singular.

Hence \[{\bar{A}}'\,A=I\Rightarrow A\,{\bar{A}}'=I\]

(10) Periodic matrix : A matrix A will be called a periodic matrix if \[{{A}^{k+1}}=A\]where k is a positive integer. If, however k is the least positive integer for which \[{{A}^{k+1}}=A,\]then k is said to be the period of A.

(11) Differentiation of a matrix : If \[A=\left[ \begin{matrix} f(x) & g(x)  \\ h(x) & l(x)  \\ \end{matrix} \right]\] then \[\frac{dA}{dx}=\left[ \begin{matrix} {f}'(x) & {g}'(x)  \\ {h}'(x) & {l}'(x)  \\ \end{matrix} \right]\]is a differentiation of matrix A.

Example : If \[A=\left[ \begin{matrix} {{x}^{2}} & \sin x  \\ 2x & 2  \\ \end{matrix} \right]\]then \[\frac{dA}{dx}=\left[ \begin{matrix} 2x & \cos x  \\ 2 & 0  \\ \end{matrix} \right]\]

(12) Conjugate of a matrix : The matrix obtained from any given matrix A containing complex number as its elements, on replacing its elements by the corresponding conjugate complex numbers is called conjugate of A and is denoted by \[\bar{A}\] .

Example: \[A=\left[ \begin{matrix} 1+2i & 2-3i & 3+4i  \\ 4-5i & 5+6i & 6-7i  \\ 8 & 7+8i & 7  \\ \end{matrix} \right]\]  then \[\bar{A}=\left[ \begin{matrix} 1-2i & 2+3i & 3-4i  \\ 4+5i & 5-6i & 6+7i  \\ 8 & 7-8i & 7  \\ \end{matrix} \right]\]\[\]

Properties of conjugates

(i) \[\overline{\left( {\bar{A}} \right)}=A\]

(ii) \[\overline{\left( A+B \right)}=\bar{A}+\bar{B}\]

(iii) \[\overline{(\alpha A)}=\bar{\alpha }\bar{A},\alpha \]being any number

(iv) \[(\overline{AB)}=\bar{A}\,\bar{B},A\]and B being conformable for multiplication

(13) Transpose conjugate of a matrix : The transpose of the conjugate of a matrix A is called transposed conjugate of A and is denoted by \[{{A}^{\theta }}.\]The conjugate of the transpose of A is the same as the transpose of the conjugate of A i.e. \[\overline{({A}')}=(\bar{A}{)}'\,={{A}^{\theta }}\].

If \[A={{[{{a}_{ij}}]}_{m\times n}}\] then \[{{A}^{\theta }}={{[{{b}_{ji}}]}_{n\times m}}\] where \[{{b}_{ji}}={{\bar{a}}_{ij}}\]

i.e., the \[{{(j,i)}^{th}}\]element of \[{{A}^{\theta }}=\] the conjugate of \[{{(i,\text{ }j)}^{th}}\] element of A.

Example: If \[A=\left[ \begin{matrix} 1+2i & 2-3i & 3+4i  \\ 4-5i & 5+6i & 6-7i  \\ 8 & 7+8i & 7  \\ \end{matrix} \right]\] then \[{{A}^{\theta }}=\left[ \begin{matrix} 1-2i & 4+5i & 8  \\ 2+3i & 5-6i & 7-8i  \\ 3-4i & 6+7i & 7  \\ \end{matrix} \right]\]

Properties of transpose conjugate

(i) \[{{({{A}^{\theta }})}^{\theta }}=A\]

(ii) \[{{(A+B)}^{\theta }}={{A}^{\theta }}+{{B}^{\theta }}\]

(iii) \[{{(kA)}^{\theta }}=\bar{K}{{A}^{\theta }},\]K being any number

(iv) \[{{(AB)}^{\theta }}={{B}^{\theta }}{{A}^{\theta }}\]