Matrices:

A rectangular arrangement of numbers is rows & columns is called a matrix.

A matrix of order mxn contains mn elements. A matrix of order mxn is of the form

$A=\left[\begin{array}{cccc}{a}_{11}& a_{12}& \dots \dots \dots & a_{1n}\\ a_{21}& a_{22}& \dots \dots \dots & a_{2n}\\ ⋮& & & \\ a_{m1}& a_{m2}& \dots \dots ..& a_{mn}\end{array}\right]={\left[a_{ij}\right]}_{m\times n}$

If $m=1$, the matrix is a row matrix. If $n=1$ the matrix is a column matrix.

Equality: If $A\mathrm{\&}B$ are of same order, then $A=B$ if $a_{ij}=b_{ij}\mathrm{\forall }i\mathrm{\&}j$

Types of matrices:

• Null (zero) matrix
• Square matrix
• Diagonal matrix

$(\text{ If }\mathrm{A}=\mathrm{diag}({\mathrm{d}}_1,{\mathrm{d}}_2\dots \dots .{\mathrm{d}}_{\mathrm{n}})\text{, then }{\mathrm{A}}^{\mathrm{n}}=({\mathrm{d}}_1^{\mathrm{n}},{\mathrm{d}}_2^{\mathrm{n}}\dots \dots .{\mathrm{d}}_{\mathrm{n}}^{\mathrm{n}}))$

• Identity matrix
• Triangular matrix
• (Determinant of upper triangular or lower triangular matrix is the product of principal diagonal elements). Also minimum number of zeros in a triangular matrix is given by ${\displaystyle \frac{\mathrm{n}(\mathrm{n}-1)}{2}}$ where ’ $n$ ’ is the order of matrix.

Properties of Matrix Multiplication

i. $\mathrm{AB}\ne \mathrm{BA}$

ii. $\mathrm{A}(\mathrm{BC})=(\mathrm{AB})\mathrm{C}$

iii. $A(B+C)=AB+AC$

iv. $\mathrm{AI}=\mathrm{A}=\mathrm{IA}$

v. $\mathrm{AB}=\mathrm{AC}$ need not imply $\mathrm{B}=\mathrm{C}$

vi. $\mathrm{AB}=0$ need not imply $\mathrm{A}=0$ or $\mathrm{B}=0$

vii. ${\mathrm{I}}^2={\mathrm{I}}^3$ ${\mathrm{I}}^{\mathrm{m}}=\mathrm{I}(\mathrm{m}\in \mathrm{z})$

Trace (spur) a Matrix

Sum of diagonal elements of a square matrix is called the trace of matrix A.

i.e. $\mathrm{tr}A=\sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{a}}_{\mathrm{ii}}={\mathrm{a}}_{11}+{\mathrm{a}}_{22}+\dots \dots ..+{\mathrm{a}}_{\mathrm{nn}}$

Trace of a skew symmetric matrix is zero.

Properties:

Let $A={\left[a_{ij}\right]}_{\text{nxn }},B={\left[b_{ij}\right]}_{\text{nxn }}\mathrm{\&}\lambda$ is a scalar

• $\mathrm{tr}(\mathrm{A}\pm \mathrm{B})=\mathrm{tr}(\mathrm{A})\pm \mathrm{tr}(\mathrm{B})$

• $\mathrm{tr}(AB)=\mathrm{tr}(BA)(\mathrm{tr}(AB)\ne \mathrm{tr}(A)\cdot \mathrm{tr}(B))$

• $\mathrm{tr}\left({\mathrm{A}}^{\mathrm{T}}\right)=\mathrm{tr}(\mathrm{A})$

• $\mathrm{tr}(\lambda \mathrm{A})=\lambda \mathrm{tr}(\mathrm{A})$

• $\mathrm{tr}\left({\mathrm{I}}_{\mathrm{n}}\right)=\mathrm{n}$

Transpose of a Matrix

the matrix obtained by interchanging the rows and columns of the given matrix a is called transpose of A.

If $A$ is of order mxn, then ${\mathrm{A}}^{\mathrm{T}}$ is of order nxm.

Properties:

• ${\left(A^T\right)}^T=A$
• $(A\pm B{)}^T=A^T\pm B^T$
• $(\lambda \mathrm{A}{)}^{\mathrm{T}}=\lambda {\mathrm{A}}^{\mathrm{T}}$, where $\lambda$ is a scalar
• $(\mathrm{AB}{)}^{\mathrm{T}}={\mathrm{B}}^{\mathrm{T}}{\mathrm{A}}^{\mathrm{T}}$ (reversal law of transposes) (if $\mathrm{A}\mathrm{\&}\mathrm{B}$ are conformable for multiplication) Also, $(\mathrm{ABC}{)}^{\mathrm{T}}={\mathrm{C}}^{\mathrm{T}}{\mathrm{B}}^{\mathrm{T}}{\mathrm{A}}^{\mathrm{T}}$ etc.
• ${\mathrm{I}}^{\mathrm{T}}=\mathrm{I}$

Conjugate of a Matrix

Conjugate of a matrix A is obtained by replacing the elements of A by their corresponding complex conjugates. It is denoted by $\overline{A}$.

Properties:

• $(\bar{\mathrm{A}})=\mathrm{A}$

• $(\overline{\mathrm{A}+\mathrm{B}})=\bar{\mathrm{A}}+\bar{\mathrm{B}}$

• $(\overline{\mathrm{AB}})=\bar{\mathrm{A}}\cdot \bar{\mathrm{B}}$

• $(\overline{\mathrm{kA}})=\bar{\mathrm{k}}\cdot \bar{\mathrm{A}}$

• $\left(\overline{{\mathrm{A}}^{\mathrm{n}}}\right)=(\bar{\mathrm{A}}{)}^{\mathrm{n}}$

Tranjugate (Transposed conjugate of a Matrix)

Transposed conjugate is obtained by interchanging the rows and columns of the matrix obtained by replacing the elements of A by their corresponding complex conjugate. It is denoted by ${\mathrm{A}}^{\ast }$.

Properties:

• ${\left({\mathrm{A}}^{\ast }\right)}^{\ast }=\mathrm{A}$
• $(A+B{)}^{\ast }=A^{\ast }+B^{\ast }$
• $(\mathrm{AB}{)}^{\ast }={\mathrm{B}}^{\ast }{\mathrm{A}}^{\ast }$
• $(\mathrm{kA}{)}^{\ast }=\bar{\mathrm{k}}{\mathrm{A}}^{\ast }$
• ${\left(A^n\right)}^{\ast }={\left(A^{\ast }\right)}^n$

Symmetric and skew symmetric matrices:

A square matrix $\mathrm{A}$ is called symmetric if ${\mathrm{A}}^{\mathrm{T}}=\mathrm{A}$ and skew symmetric if ${\mathrm{A}}^{\mathrm{T}}=-\mathrm{A}$. All principal diagonal elements of a skew-sym metric matrix are zero.

Properties:

• If $\mathrm{A}$ is a square matrix, then $\mathrm{A}+{\mathrm{A}}^{\mathrm{T}},{\mathrm{AA}}^{\mathrm{T}},{\mathrm{A}}^{\mathrm{T}}\mathrm{A}$ are symmetric matrices, while $\mathrm{A}-{\mathrm{A}}^{\mathrm{T}}$ is skew-symmetric matrix.
• If $\mathrm{A}$ is symmetric matrix, then - $\mathrm{A},\mathrm{KA},{\mathrm{A}}^{\mathrm{T}},{\mathrm{A}}^{\mathrm{n}},{\mathrm{A}}^{-1},{\mathrm{B}}^{\mathrm{T}}\mathrm{AB}$ are also symmetric matrices where $B$ is a square matrix of same order that of $A$.
• If $A$ is a skew symmetric matrix, then $A^{2n}$ is symmetric where as $A^{2n+1}$ and $B^{T}AB$ are skew symmetric ( $n\in N\mathrm{\&}B$ is a square matrix of same order that of $A$ )
• If $A\mathrm{\&}B$ are two symmetric matrices, then $A\pm B,AB+BA$ and $AB-BA$ are skew symmetric.
• If $\mathrm{A}\mathrm{\&}\mathrm{B}$ are two skew symmetric matrices, then $\mathrm{A}\pm \mathrm{B},\mathrm{AB}-\mathrm{BA}$ are skew symmetric and $\mathrm{AB}+\mathrm{BA}$ is symmetric.
• Every square matrix can be uniquely expressed as sum of a symmetric and a skew symmetric matrix.

$\mathrm{A}={\displaystyle \frac{1}{2}}(\mathrm{A}+{\mathrm{A}}^{\mathrm{T}})+{\displaystyle \frac{1}{2}}(\mathrm{A}-{\mathrm{A}}^{\mathrm{T}})=\mathrm{P}+\mathrm{Q}$. Here $\mathrm{P}$ is symmetric and $\mathrm{Q}$ is skew symmetric.

• If $A$ is a skew symmetric matrix \& $C$ is a column matrix, then $C^{\mathrm{T}}\mathrm{AC}$ is a null matrix.

• If $A$ is a skew symmetric matrix of odd order, then ${\mathrm{A}}^{-1}$ does not exists $(\because |\mathrm{A}|=0)$

• Null matrix is both symmetric and skew symmetric

• All elements on the principal diagonal of a skew-symmetric matrix are always zero.

Hermitian Skew-Hermitian matrix

A square matrix is said to be Hermitian if $\overline{{\mathrm{A}}^{\mathrm{T}}}=\mathrm{A}$ and skew-hermitian if $\overline{{\mathrm{A}}^{\mathrm{T}}}=-\mathrm{A}$.

Properties:

The diagonal elements of a Hermitian matrix are real where that of a skew-Hermitian matrix are either purely imaginary or zero.

• Every square matrix (with complex elements) can be uniquely expressed as the sum of Hermitian and skew-Hermitian matrices.

$\mathrm{A}=\underset{\text{Hermitian }}{\underbrace{{\displaystyle \frac{1}{2}}(\mathrm{A}+\overline{{\mathrm{A}}^{\mathrm{T}}})}}+\underset{\text{Skew-hermitian }}{\underbrace{{\displaystyle \frac{1}{2}}(\mathrm{A}-\overline{{\mathrm{A}}^{\mathrm{T}}})}}$

Orthogonal matrix

• A square matrix $A$ is called on orthogonal matrix if ${\mathrm{AA}}^{\mathrm{T}}={\mathrm{A}}^{\mathrm{T}}\mathrm{A}=\mathrm{I}$.
• If $\mathrm{A}$ is orthogonal, then $|\mathrm{A}|=\pm 1$. Hence it is non-singular.
• If $A$ is orthogonal, it is invertible with ${\mathrm{A}}^{-1}={\mathrm{A}}^{\mathrm{T}}$.
• If $\mathrm{A}\mathrm{\&}\mathrm{B}$ area orthogonal matrices of order $\mathrm{n}$, then $\mathrm{AB},\mathrm{BA},{\mathrm{A}}^{-1},{\mathrm{A}}^{\mathrm{T}}$ are orthogonal.
• If $A$ is orthogonal with $|A|=1$, then each element of $A$ is equal to its cofactor is $|A|$.
• If $A$ is orthogonal with $|A|=1$, then each element of $A$ is equal to the negative of its cofactor is $|\mathrm{A}|$.
• If ${\mathrm{A}}_{3\times 3}$ is orthogonal and ${\mathrm{B}}_{3\times 3}$ is a skew symmetric matrix, then $|\mathrm{AB}|=1$.

Unitary Matrix

A square matrix $A$ is called a unitary matrix. If $A\overline{A^{\mathrm{T}}}=\overline{{\mathrm{A}}^{\mathrm{T}}}\mathrm{A}=\mathrm{I}$.

Properties:

• Determinant of a unitary matrix is of unit modulus.
• If $\mathrm{A}$ is a unitary matrix, then ${\mathrm{A}}^{\mathrm{T}},\bar{\mathrm{A}},\overline{{\mathrm{A}}^{\mathrm{T}}}$ and ${\mathrm{A}}^{-1}$ are unitary.
• Product of two unit matrices is unitary.

Idempotent matrix

A square matrix is called idempotent if ${\mathrm{A}}^2=\mathrm{A}$.

Properties:

• If $\mathrm{A}$ is idempotent, then I-A is also idempotent.
• If $\mathrm{A},\mathrm{B}$ are two idempotent matrices and $\mathrm{AB}=\mathrm{BA}=0$, then $(\mathrm{A}+\mathrm{B})$ is idempotent.
• If $\mathrm{AB}=\mathrm{A};\mathrm{BA}=\mathrm{B}$, then $\mathrm{A}\mathrm{\&}\mathrm{B}$ are idempotent matrices and ${\mathrm{A}}^{\mathrm{n}}+{\mathrm{B}}^{\mathrm{n}}=\mathrm{A}+\mathrm{B}$ where $\mathrm{n}\in \mathrm{N}$.

Periodic matrix

A square matrix $A$ is called periodic if ${\mathrm{A}}^{k+1}=\mathrm{A};\mathrm{k}\in {\mathrm{Z}}^{+}$. The least value of $\mathrm{k}$ is called period of A.

When $\mathrm{k}=1$, we get ${\mathrm{A}}^2=\mathrm{A}$ and it becomes an idempotent matrix.

Nilpotent Matrix

A square matrix $A$ is called Nilpotent of order $k$ if $A^k=0$ and $A^{k-1}\ne 0,k\in Z^{+}$. Here $k$ is called the order of the nilpotent matrix $A$.

Involutory Matrix

A square matrix $\mathrm{A}$ is called involutory if ${\mathrm{A}}^2=\mathrm{I}$.

i.e. ${\mathrm{A}}^{-1}=\mathrm{A}(\mathrm{A}$ is the inverse of itself)

Adjoint of a square matrix

The transpose of the matrix of cofactors $\mathrm{C}$ is called the adjoint of matrix $\mathrm{A}$ and is denoted by adj A.

Properties:

For square matrices $A\mathrm{\&}B$ of order $n$,

• $A(\mathrm{adj}A)=(\mathrm{adj}A)A=|A|I_n$

• $\mathrm{adj}(\mathrm{AB})=(\mathrm{adjB})(\mathrm{adj}A)$

• $\left(\mathrm{adj}{A}^T\right)=(\mathrm{adj}A{)}^{\mathrm{T}}$

• $\left(\mathrm{adj}{A}^m\right)=(\mathrm{adj}A{)}^m;m\in N$

• $\mathrm{adj}(kA)=k^{n-1}(\mathrm{adj}A);k\in R$

• Adjoint of a diagonal matrix is a diagonal matrix.

• $|\mathrm{adj}\mathrm{A}|=|\mathrm{A}{|}^{\mathrm{n}-1}$

• $\mathrm{adj}(\mathrm{adj}A)=|A{|}^{n-2}A;|A|\ne 0$

• $|\mathrm{adj}(\mathrm{adj}A)|=|A{|}^{(\mathrm{n}-1{)}^2};|\mathrm{A}|\ne 0$

Inverse of a matrix

For a non singular matrix A of order $\mathrm{n},{\mathrm{A}}^{-1}={\displaystyle \frac{1}{|\mathrm{A}|}}(\mathrm{adj}\mathrm{A})$

Properties:

• ${\left({\mathrm{A}}^{-1}\right)}^{-1}=\mathrm{A}$
• ${\left(A^T\right)}^{-1}={\left(A^{-1}\right)}^T$
• $\left(\mathrm{adj}{\mathrm{A}}^{-1}\right)=(\mathrm{adj}\mathrm{A}{)}^{-1}$
• $\left|{\mathrm{A}}^{-1}\right|={\displaystyle \frac{1}{|\mathrm{A}|}}=|\mathrm{A}{|}^{-1}$
• If $\mathrm{A}=\mathrm{diag}({\mathrm{a}}_{11},{\mathrm{a}}_{22}\dots ..{\mathrm{a}}_{\mathrm{nn}})$

${\mathrm{A}}^{-1}=\mathrm{diag}({\mathrm{a}}_{11}^{-1},{\mathrm{a}}_{22}^{-1},\dots {\mathrm{a}}_{\mathrm{nn}}^{-1})$

• $(\mathrm{AB})={\mathrm{B}}^{-1}{\mathrm{A}}^{-1}$ (reversal law)

• $\mathrm{AB}=\mathrm{AC}\Rightarrow \mathrm{A}=\mathrm{C}$ if $|\mathrm{A}|\ne 0$.

Solved Examples

1. If $A=\left[\begin{array}{ll}\alpha & 2\\ 2& \alpha \end{array}\right]$ and $\left|A^3\right|=125$, then the value of $\alpha$ is

(a) $\pm 1$

(b) $\pm 2$

(c) $\pm 3$

(d) $\pm 5$

2. If $\mathrm{A}=\left[\begin{array}{ll}1& 2\\ 3& 4\end{array}\right]$, then $\mathrm{I}+\mathrm{A}+{\mathrm{A}}^2+{\mathrm{A}}^3+—-\mathrm{\infty }$ equals to

(a) $\left[\begin{array}{ll}1& 0\\ 0& 1\end{array}\right]$

(b) $\left[\begin{array}{rr}-1& -2\\ -3& -4\end{array}\right]$

(c) $\left[\begin{array}{cc}{\displaystyle \frac{1}{2}}& -{\displaystyle \frac{1}{3}}\\ -{\displaystyle \frac{1}{2}}& 0\end{array}\right]$

(d) none of these

3. If $\mathrm{A}$ is non-singular and $(\mathrm{A}-2\mathrm{I})(\mathrm{A}-4\mathrm{I})=0$, then ${\displaystyle \frac{1}{6}}\mathrm{A}+{\displaystyle \frac{4}{3}}{\mathrm{A}}^{-1}=$

(a) $\mathrm{I}$

(b) 0

(c) $2\mathrm{I}$

(d) $6\mathrm{I}$

4. If $A$ and $B$ are square matrices such that $B=-A^{-1}BA$, then

(a) $AB+BA=0$

(b) $(A+B{)}^2=A^2+B^2$

(c) $(A+B{)}^2=A^2+2AB+B^2$

(d) $(A+B{)}^2=A+B$

5. If $A={\displaystyle \frac{1}{3}}\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& -2\\ \mathrm{a}& 2& \mathrm{b}\end{array}\right]$ is an orthogonal matrix, then

(a) $\mathrm{a}=2,\mathrm{b}=1$

(b) $\mathrm{a}=-2,\mathrm{b}=-1$

(c) $\mathrm{a}=2,\mathrm{b}=-1$

(d) $\mathrm{a}=-2,\mathrm{b}=1$

6. If $\mathrm{P}=\left[\begin{array}{cc}{\displaystyle \frac{\sqrt{3}}{2}}& {\displaystyle \frac{1}{2}}\\ {\displaystyle \frac{-1}{2}}& {\displaystyle \frac{\sqrt{3}}{2}}\end{array}\right],\mathrm{A}=\left[\begin{array}{ll}1& 1\\ 0& 1\end{array}\right]$ and $\mathrm{Q}={\mathrm{PAP}}^{\mathrm{T}}$ then ${\mathrm{P}}^{\mathrm{T}}{\mathrm{Q}}^{2005}\mathrm{P}=$

(a) $\left[\begin{array}{cc}1& 2005\\ 0& 1\end{array}\right]$

(b) $\left[\begin{array}{cc}4+2005\sqrt{3}& 6015\\ 2005& 4-2005\sqrt{3}\end{array}\right]$

(c) ${\displaystyle \frac{1}{4}}\left[\begin{array}{cc}1& 2005\\ 0& 1\end{array}\right]$

(d) none of these

7. If $A=\left[\begin{array}{cc}2& 1\\ -4& -2\end{array}\right]$, then the value of $I+2A+3A^2+——–\mathrm{\infty }\mathrm{\infty }$ is

(a) $\left[\begin{array}{cc}4& 1\\ -4& 0\end{array}\right]$

(b) $\left[\begin{array}{cc}3& 1\\ -4& -1\end{array}\right]$

(c) $\left[\begin{array}{cc}5& 2\\ -8& -3\end{array}\right]$

(d) none of these

Exercise

1. Consider an arbitrary $3\times 3$ matrix $A=\left[a_{ij}\right]$, a matrix $B=\left[b_{ij}\right]$ is formed such that $b_{ij}$ is the sum of all the elements except $a_{\mathrm{ij}}$ in the ${\mathrm{i}}_{\text{th }}$ row of $A$. If there exists a matrix $X$ with constant elements such that $\mathrm{AX}=\mathrm{B}$, then $\mathrm{X}$ is

(a) skew symmetric

(b) null matrix

(c) diagonal matrix

(d) none of these

2. Let A be a square matrix all of whose entries are integers. Then which one of the following is true?

(a) If $|\mathrm{A}|=\pm 1$, then ${\mathrm{A}}^{-1}$ exists but all its entries are not necessarily integers.

(b) If $|A|=\pm 1$, then ${\mathrm{A}}^{-1}$ exists and all its entries are non integers.

(c) If $|\mathrm{A}|=\pm 1$, then ${\mathrm{A}}^{-1}$ exists and all its entries are integers.

(d) If $|A|=\pm 1$, then $A^{-1}$ need not exist.

3. $\mathrm{X},\mathrm{Y}\mathrm{\&}\mathrm{Z}$ are positive numbers greater than 10 such, that $\mathrm{Y}$ and $\mathrm{Z}$ have respectively $1\mathrm{\&}0$ at their unit’s place and $\mathrm{\Delta }$ is the determinant $\left|\begin{array}{lll}\mathrm{X}& 4& 1\\ \mathrm{Y}& 0& 1\\ \mathrm{Z}& 1& 0\end{array}\right|$. If $(\mathrm{\Delta }+1)$ is divisible by 10 then $\mathrm{X}$ has its unit’s place

(a) 1

(b) 0

(c) 2

(d) none of these

4. If $\mathrm{P}$ is non singular matrix, then value of $\mathrm{adj}\left({\mathrm{P}}^{-1}\right)$ in terms of $\mathrm{P}$ is

(a) ${\displaystyle \frac{\mathrm{P}}{|\mathrm{P}|}}$

(b) $\mathrm{P}|\mathrm{P}|$

(c) $\mathrm{P}$

(d) none of these

5. If $\mathrm{A}=\left[\begin{array}{ll}\mathrm{a}& \mathrm{b}\\ 0& \mathrm{a}\end{array}\right]$ is ${\mathrm{n}}^{\text{th }}$ root of ${\mathrm{I}}_2$ then choose the correct statements.

i. if $\mathrm{n}$ is odd, $\mathrm{a}=1,\mathrm{b}=0$

ii. if $\mathrm{n}$ is odd, $\mathrm{a}=-1,\mathrm{b}=0$

iii. if $\mathrm{n}$ is even, $\mathrm{a}=1,\mathrm{b}=0$

iv. if $\mathrm{n}$ is even $\mathrm{a}=-1,\mathrm{b}=0$

(a) i, ii, iii

(b) ii, iii, iv

(c) i,ii,iii

(d) i, iii,iv

6. If $A^2=I$, then the value of $|A-I|$ (where $A$ has order 3$)$

(a) 1

(b) -1

(c) 0

(d) cannot say anything

7. If $A=\left[\begin{array}{ll}1& 2\\ 2& 1\end{array}\right]\mathrm{\&}f(x)={\displaystyle \frac{1+x}{1-x}}$ then $f(A)$ is

(a) $\left[\begin{array}{ll}1& 1\\ 1& 1\end{array}\right]$

(b) $\left[\begin{array}{ll}2& 2\\ 2& 2\end{array}\right]$

(c) $\left[\begin{array}{ll}-1& -1\\ -1& -1\end{array}\right]$

(d) none of these

8. If $\mathrm{f}(\mathrm{x})$ satisfies the $\left|\begin{array}{ccc}\mathrm{f}(\mathrm{x}-3)& \mathrm{f}(\mathrm{x}+4)& \mathrm{f}((\mathrm{x}+1)(\mathrm{x}-2)-(\mathrm{x}-1{)}^2)\\ 5& 4& -5\\ 5& 6& 15\end{array}\right|=0\mathrm{\forall }$ equation real $\mathrm{x}$, then

(a) $\mathrm{f}(\mathrm{x})$ is not periodic

(b) $\mathrm{f}(\mathrm{x})$ is periodic and of period 1

(c) $\mathrm{f}(\mathrm{x})$ is periodic of period 7

(d) $\mathrm{f}(\mathrm{x})$ is an odd function

9. Let $\mathrm{M}\mathrm{\&}\mathrm{N}$ be two $3\times 3$ non-singular skew-symmetric matrices such that $\mathrm{MN}=\mathrm{NM}$. If ${\mathrm{P}}^{\mathrm{T}}$ denotes the transpose of $\mathrm{P}$, then ${\mathrm{M}}^2{\mathrm{N}}^2(\mathrm{MN}{)}^{-1}{\left({\mathrm{M}}^{\mathrm{T}}{\mathrm{N}}^{-1}\right)}^{\mathrm{T}}$ is equals

(a) ${\mathrm{M}}^2$

(b) $-{\mathrm{N}}^2$

(c) $-{\mathrm{M}}^2$

(d) $\mathrm{MN}$

10. Read the following and answer the questions.

Let $\mathrm{p}$ be an odd prime number and ${\mathrm{T}}_{\mathrm{p}}$ be the following set of $2\times 2$ matrices ${\mathrm{T}}_{\mathrm{P}}=\{\mathrm{A}=\left[\begin{array}{ll}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{a}\end{array}\right];\mathrm{a},\mathrm{b},\mathrm{c}\in \{0,1,2,\dots .\mathrm{p}-1\}\}$.

i. The number $\mathrm{A}$ is ${\mathrm{T}}_{\mathrm{p}}$ such that $\mathrm{A}$ is either symmetric or skew-symmetric or both, and $\det (\mathrm{A})$ is divisible by $\mathrm{p}$ is

(a) $(\mathrm{p}-1{)}^2$

(b) $2(\mathrm{p}-1)$

(c) $(\mathrm{p}-1{)}^2+1$

(d) $2\mathrm{p}-1$

ii. The number of $A$ in $T_p$ such that the trace of $A$ is not divisible by $p$ but det $(A)$ is divisible by $\mathrm{p}$ is

(a) $(\mathrm{p}-1)({\mathrm{p}}^2-\mathrm{p}+1)$

(b) ${\mathrm{p}}^3-(\mathrm{p}-1{)}^2$

(c) $(\mathrm{p}-1{)}^2$

(d) $(p-1)(p^2-2)$

iii. The number of $A$ is $T_p$ such that $\det (\mathrm{A})$ is not divisible by $\mathrm{p}$ is

(a) $2{\mathrm{p}}^2$

(b) ${\mathrm{p}}^3-5\mathrm{p}$

(c) ${\mathrm{p}}^3-3\mathrm{p}$

(d) $p^3-p^2$

11.* An item of column I can be matched with more than one item of column II. All the items of column II are to be matched

12. In $\mathrm{△}\mathrm{ABC}$, if $\left|\begin{array}{lll}1& \mathrm{a}& \mathrm{b}\\ 1& \mathrm{c}& \mathrm{a}\\ 1& \mathrm{b}& \mathrm{c}\end{array}\right|=0$, then the value of $({\sin }^2\mathrm{A}+{\sin }^2\mathrm{B}+{\sin }^2\mathrm{C})64$ must be

(a) 64

(b) -64

(c) 144

(d) none of these