# JEE Main & Advanced Mathematics Matrices Special Types of Matrices

## Special Types of Matrices

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) ${{(AB)}^{\theta }}={{B}^{\theta }}{{A}^{\theta }}$

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