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

$A=\left[\begin{array}{cccc} a _{11} & a _{12} & \ldots \ldots \ldots & a _{1 n} \\ a _{21} & a _{22} & \ldots \ldots \ldots & a _{2 n} \\ \vdots & & & \\ a _{m 1} & a _{m 2} & \ldots \ldots . . & a _{m n} \end{array}\right]=\left[a _{i j}\right] _{m \times n}$

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

Equality: If $A \& B$ are of same order, then $A=B$ if $a _{i j}=b _{i j} \forall i \& j$

Types of matrices:

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

$\hspace {0.7 cm}\left(\text { If } \mathrm{A}=\operatorname{diag}\left(\mathrm{d} _{1}, \mathrm{~d} _{2} \ldots \ldots . \mathrm{d} _{\mathrm{n}}\right) \text {, then } \mathrm{A}^{\mathrm{n}}=\left(\mathrm{d} _{1}^{\mathrm{n}}, \mathrm{d} _{2}^{\mathrm{n}} \ldots \ldots . \mathrm{d} _{\mathrm{n}}^{\mathrm{n}}\right)\right)$

• 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 $\dfrac{\mathrm{n}(\mathrm{n}-1)}{2}$ where ’ $n$ ’ is the order of matrix.

Properties of Matrix Multiplication

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

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

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

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

Trace of a skew symmetric matrix is zero.

Properties:

Let $A=\left[a _{i j}\right] _{\text {nxn }}, B=\left[b _{i j}\right] _{\text {nxn }} \& \lambda$ is a scalar

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

• $\operatorname{tr}(A B)=\operatorname{tr}(B A)(\operatorname{tr}(A B) \neq \operatorname{tr}(A) \cdot \operatorname{tr}(B))$

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

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

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

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 $\mathrm{A} \& \mathrm{~B}$ 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 $\bar{A}$.

Properties:

• $(\overline{\mathrm{A}})=\mathrm{A}$

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

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

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

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

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 $\mathrm{A}^{*}$.

Properties:

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

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 $A^{2 n}$ is symmetric where as $A^{2 n+1}$ and $B^{T} A B$ are skew symmetric ( $n \in N \& B$ is a square matrix of same order that of $A$ )
• If $A \& B$ are two symmetric matrices, then $A \pm B, A B+B A$ and $A B-B A$ are skew symmetric.
• If $\mathrm{A} \& \mathrm{~B}$ 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.

$\hspace {0.7 cm}\mathrm{A}=\dfrac{1}{2}\left(\mathrm{~A}+\mathrm{A}^{\mathrm{T}}\right)+\dfrac{1}{2}\left(\mathrm{~A}-\mathrm{A}^{\mathrm{T}}\right)=\mathrm{P}+\mathrm{Q}$. 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.

$\hspace {1 cm}\mathrm{A}=\underbrace{\dfrac{1}{2}\left(\mathrm{~A}+\overline{\mathrm{A}^{\mathrm{T}}}\right)} _{\text {Hermitian }}+\underbrace{\dfrac{1}{2}\left(\mathrm{~A}-\overline{\mathrm{A}^{\mathrm{T}}}\right)} _{\text {Skew-hermitian }}$

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 $\mathrm{A} \& \mathrm{~B}$ 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 $\mathrm{A}^{\mathrm{T}}, \overline{\mathrm{A}}, \overline{\mathrm{A}^{\mathrm{T}}}$ 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 $\mathrm{A} \& \mathrm{~B}$ 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 $A^{k-1} \neq 0, k \in Z^{+}$. 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 $A \& B$ of order $n$,

• $A(\operatorname{adj} A)=(\operatorname{adj} A) A=|A| I _{n}$

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

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

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

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

• Adjoint of a diagonal matrix is a diagonal matrix.

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

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

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

Inverse of a matrix

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

Properties:

• $\left(\mathrm{A}^{-1}\right)^{-1}=\mathrm{A}$
• $\left(A^{T}\right)^{-1}=\left(A^{-1}\right)^{T}$
• $\left(\operatorname{adj} \mathrm{A}^{-1}\right)=(\operatorname{adj} \mathrm{A})^{-1}$
• $\left|\mathrm{A}^{-1}\right|=\dfrac{1}{|\mathrm{~A}|}=|\mathrm{A}|^{-1}$
• If $\mathrm{A}=\operatorname{diag}\left(\mathrm{a} _{11}, \mathrm{a} _{22} \ldots . . \mathrm{a} _{\mathrm{nn}}\right)$

$\hspace {0.7 cm}\mathrm{A}^{-1}=\operatorname{diag}\left(\mathrm{a} _{11}^{-1}, \mathrm{a} _{22}^{-1}, \ldots \mathrm{a} _{\mathrm{nn}}^{-1}\right)$

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

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

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 $\mathrm{I}+\mathrm{A}+\mathrm{A}^{2}+\mathrm{A}^{3}+—-\infty$ 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) $\left[\begin{array}{cc}\dfrac{1}{2} & -\dfrac{1}{3} \\ -\dfrac{1}{2} & 0\end{array}\right]$

(d) none of these

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

(a) $\mathrm{I}$

(b) 0

(c) $2 \mathrm{I}$

(d) $6 \mathrm{I}$

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

(a) $A B+B A=0$

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

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

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

5. If $A=\dfrac{1}{3}\left[\begin{array}{ccc}1 & 2 & 2 \\ 2 & 1 & -2 \\ \mathrm{a} & 2 & \mathrm{~b}\end{array}\right]$ 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 $\mathrm{P}=\left[\begin{array}{cc}\dfrac{\sqrt{3}}{2} & \dfrac{1}{2} \\ \dfrac{-1}{2} & \dfrac{\sqrt{3}}{2}\end{array}\right], \mathrm{A}=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$ 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) $\dfrac{1}{4}\left[\begin{array}{cc}1 & 2005 \\ 0 & 1\end{array}\right]$

(d) none of these

7. If $A=\left[\begin{array}{cc}2 & 1 \\ -4 & -2\end{array}\right]$, then the value of $I+2 A+3 A^{2}+——–\infty \infty$ 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 $A=\left[a _{i j}\right]$, a matrix $B=\left[b _{i j}\right]$ is formed such that $b _{i j}$ 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. $\mathrm{X}, \mathrm{Y} \& \mathrm{Z}$ are positive numbers greater than 10 such, that $\mathrm{Y}$ and $\mathrm{Z}$ have respectively $1 \& 0$ at their unit’s place and $\Delta$ is the determinant $\left|\begin{array}{lll}\mathrm{X} & 4 & 1 \\ \mathrm{Y} & 0 & 1 \\ \mathrm{Z} & 1 & 0\end{array}\right|$. If $(\Delta+1)$ 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 $\operatorname{adj}\left(\mathrm{P}^{-1}\right)$ in terms of $\mathrm{P}$ is

(a) $\dfrac{\mathrm{P}}{|\mathrm{P}|}$

(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 $A=\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right] \& f(x)=\dfrac{1+x}{1-x}$ 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 $\left|\begin{array}{ccc}\mathrm{f}(\mathrm{x}-3) & \mathrm{f}(\mathrm{x}+4) & \mathrm{f}\left((\mathrm{x}+1)(\mathrm{x}-2)-(\mathrm{x}-1)^{2}\right) \\ 5 & 4 & -5 \\ 5 & 6 & 15\end{array}\right|=0 \forall$ 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 $\mathrm{M} \& \mathrm{~N}$ 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 $\mathrm{T} _{\mathrm{P}}=\left\{\mathrm{A}=\left[\begin{array}{ll}\mathrm{a} & \mathrm{b} \\ \mathrm{c} & \mathrm{a}\end{array}\right] ; \mathrm{a}, \mathrm{b}, \mathrm{c} \in\{0,1,2, \ldots . \mathrm{p}-1\}\right\}$.

i. The number $\mathrm{A}$ is $\mathrm{T} _{\mathrm{p}}$ such that $\mathrm{A}$ is either symmetric or skew-symmetric or both, and $\operatorname{det}(\mathrm{A})$ 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) $(\mathrm{p}-1)\left(\mathrm{p}^{2}-\mathrm{p}+1\right)$

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

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

(d) $(p-1)\left(p^{2}-2\right)$

iii. The number of $A$ is $T _{p}$ such that $\operatorname{det}(\mathrm{A})$ 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 $\triangle \mathrm{ABC}$, 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 $\left(\sin ^{2} \mathrm{~A}+\sin ^{2} \mathrm{~B}+\sin ^{2} \mathrm{C}\right) 64$ must be

(a) 64

(b) -64

(c) 144

(d) none of these