Matrices - Notes, Concept and All Important Formula

MATRICES

1. INTRODUCTION:

A rectangular array of mn numbers in the form of $m$ horizontal lines (called rows) and $n$ vertical lines (called columns), is called a matrix of order $m$ by $n$, written as $m \times n$ matrix. In compact form, the matrix is represented by $\mathrm{A}=\left[\mathrm{a}_{\mathrm{i}}\right]_{\mathrm{mx}}$

---

All Chapter Notes, Concept and Important Formula

---

2. SPECIAL TYPE OF MATRICES :

(a) Row Matrix (Row vector) : $A=\left[a_{11}, a_{12}, \ldots \ldots \ldots . a_{1 n}\right]$ i.e. row matrix has exactly one row.
(b) Column Matrix (Column vector) : $A=\left[\begin{array}{c}a_{11} \\ a_{21} \\ : \\ a_{m 1}\end{array}\right]$ i.e. columr matrix has exactly one column.
(c) Zero or Null Matrix : $\left(A=O_{m \times n}\right)$, An $m \times n$ matrix whose all entries are zero.
(d) Horizontal Matrix : A matrix of order $\mathrm{m} \times \mathrm{n}$ is a horizontal matrix if $\mathrm{n}>\mathrm{m}$.
(e) Vertical Matrix : A matrix of order $m \times n$ is a vertical matrix if $\mathrm{m}>\mathrm{n}$
(f) Square Matrix : (Order n) If number of rows $=$ number of column, then matrix is a square matrix.

Note :
(i) The pair of elements $a_{i j}$ & $a_{\text {in }}$ are called Conjugate Elements.
(ii) The elements $\mathrm{a}_{11}, \mathrm{a}_{22}, \mathrm{a}_{33}, \ldots \ldots . \mathrm{a}_{\mathrm{m}}$ are called Diagonal Elements. The line along which the diagonal elements lie is called "Principal or Leading diagonal". The quantity $\sum \mathrm{a}_{\mathrm{ii}}=$ trace of the matrix written as, $\mathrm{t}_{\mathrm{r}}(\mathrm{A})$

---

---

3. SQUARE MATRICES :

Note :
(i) Minimum number of zeros in triangular matrix of order $\mathrm{n}$ $=\mathrm{n}(\mathrm{n}-1) / 2$
(ii) Minimum number of zeros in a diagonal matrix of order $\mathrm{n}$ $=\mathrm{n}(\mathrm{n}-1)$
(iii) Null square matrix is also a diagonal matrix.

---

---

4. EQUALITY OF MATRICES :

Matrices $A=\left[a_{i_{i}}\right]$ & $B=\left[b_{i j}\right]$ are equal if,
(a) both have the same order.
(b) $\mathrm{a}_{\mathrm{ij}}=\mathrm{b}_{\mathrm{iij}}$ for each pair of $\mathrm{i}$ & $\mathrm{j}$.

---

---

5. ALGEBRA OF MATRICES :

(I) Addition : $A+B=\left[a_{i j}+b_{i j}\right]$ where $A$ & $B$ are of the same order.

(a) Addition of matrices is commutative: $A+B=B+A$
(b) Matrix addition is associative: $(A+B)+C=A+(B+C)$
(c) $\mathrm{A}+\mathrm{O}=\mathrm{O}+\mathrm{A}$ (Additive identity)
(d) $A+(-A)=(-A)+A=O$ (Additive inverse)

(II) Multiplication of A Matrix By A Scalar:

If $A=\left[\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right]$, then $k A=\left[\begin{array}{lll}k a & k b & k c \\ k b & k c & k a \\ k c & k a & k b\end{array}\right]$

(III) Multiplication of matrices (Row by Column):

Let A be a matrix of order $m \times n$ and $B$ be a matrix of order $p$ $\times \mathrm{q}$ then the matrix multiplication $\mathrm{AB}$ is possible if and only if $\mathrm{n}=\mathrm{p}$

Let $A_{m \times n}=\left[a_{i j}\right]$ and $B_{n \times p}=\left[b_{i j}\right]$, then order of $A B$ is $m \times p$ & $\boxed{(\mathrm{AB})_{\mathrm{ij}}=\sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{a}_{\mathrm{ir}} \mathrm{b}_{\mathrm{rj}}}$

(IV)Properties of Matrix Multiplication:

(a) $A B=O \neq A=O$ or $B=O$ (in general)
Note :
If $A$ and $B$ are two non-zero matrices such that $A B=O$, then $A$ and $B$ are called the divisors of zero. If $A$ and $B$ are two matrices such that
(i) $A B=B A$ then $A$ and $B$ are said to commute
(ii) $\mathrm{AB}=-\mathrm{BA}$ then $\mathrm{A}$ and $\mathrm{B}$ are said to anticommute

(b) Matrix Multiplication Is Associative :
If $A, B$ & $C$ are conformable for the product $A B$ & $B C$, then $(\mathrm{AB}) \mathrm{C}=\mathrm{A}(\mathrm{BC})$
(c) Distributivity :
$A(B+C)=A B+A C]$ Provided $A, B$ & $C$ are conformable $(A+B) C=A C+B C$ for respective products

(V) Positive Integral Powers of A square matrix :

(a) $A^{m} A^{n}=A^{m+n}$
(b) $\left(\mathrm{A}^{\mathrm{m}}\right)^{\mathrm{n}}=\mathrm{A}^{\mathrm{mn}}=\left(\mathrm{A}^{\mathrm{n}}\right)^{\mathrm{m}}$
(c) $\mathrm{I}^{\mathrm{m}}=\mathrm{I}, \mathrm{m}, \mathrm{n} \in \mathrm{N}$

---

---

6. CHARACTERISTIC EQUATION:

Let $A$ be a square matrix. Then the polynomial in $x,|A-x I|$ is called as characteristic polynomial of $A$ $\&$ the equation $|A-x I|=0$ is called characteristic equation of $A$

---

---

7. CAYLEY - HAMILTON THEOREM :

Every square matrix A satisfy its characteristic equation i.e. $a_{0} x^{n}+a_{1} x^{n-1}+\ldots \ldots . .+a_{n-1} x+a_{n}=0$ is the characteristic equation of matrix $A$, then $a_{0} A^{n}+a_{1} A^{n-1}+\ldots \ldots . .+a_{n-1} A+a_{n} I=0$

---

---

8. TRANSPOSE OF A MATRIX : (Changing rows & columns)

Let $A$ be any matrix of order $m \times n$. Then transpose of $A$ is $A^{T}$ or $A^{\prime}$ of order $\mathrm{n} \times \mathrm{m}$ and $\left(\mathrm{A}^{\mathrm{T}}\right)_{\mathrm{ij}}=(\mathrm{A}_{ji})$.

Properties of transpose :

If $A^{\mathrm{T}}$ & $\mathrm{~B}^{\mathrm{T}}$ denote the transpose of $\mathrm{A}$ and $\mathrm{B}$
(a) $(\mathrm{A}+\mathrm{B})^{\mathrm{T}}=\mathrm{A}^{\mathrm{T}}+\mathrm{B}^{\mathrm{T}} ;$ note that $\mathrm{A}$ & $\mathrm{~B}$ have the same order.
(b) $(\mathrm{A} \mathrm{B})^{\mathrm{T}}=\mathrm{B}^{\mathrm{T}} \mathrm{A}^{\mathrm{T}}$ (Reversal law) $\mathrm{A}$ & $\mathrm{~B}$ are conformable for matrix product $A B$
(c) $\left(\mathrm{A}^{\mathrm{T}}\right)^{\mathrm{T}}=\mathrm{A}$
(d) $(\mathrm{kA})^{\mathrm{T}}=\mathrm{kA}^{\mathrm{T}}$, where $\mathrm{k}$ is a scalar.
General: $\left(A_{1} \cdot A_{2}, \ldots \ldots . A_{n}\right)^{\mathrm{T}}=A_{n}^{\mathrm{T}} \cdots \cdots \cdot \cdot A_{2}^{\mathrm{T}} \cdot A_{1}^{\mathrm{T}}$ (reversal law for transpose)

---

---

9. ORTHOGONAL MATRIX

A square matrix is said to be orthogonal matrix if $\mathrm{A} \mathrm{A}^{\mathrm{T}}=\mathrm{I}$

Note :
(i) The determinant value of orthogonal matrix is either 1 or $-1$. Hence orthogonal matrix is always invertible
(ii) $\mathrm{AA}^{\mathrm{T}}=\mathrm{I}=\mathrm{A}^{\mathrm{T}} \mathrm{A}$. Hence $\mathrm{A}^{-1}=\mathrm{A}^{\mathrm{T}}$.

---

---

10. SOME SPECIAL SQUARE MATRICES :

(a) Idempotent Matrix : A square matrix is idempotent provided $\mathrm{A}^{2}=\mathrm{A} .$

For idempotent matrix note the following:
(i) $A^{n}=A\,\,  \forall \,\, n \in N$
(ii) determinant value of idempotent matrix is either 0 or 1
(iii) If idempotent matrix is invertible then it will be an identity matrix i.e. I.

(b) Periodic Matrix : A square matrix which satisfies the relation $\mathrm{A}^{\mathrm{k}+1}=\mathrm{A}$, for some positive integer $\mathrm{K}$, is a periodic matrix. The period of the matrix is the least value of $\mathrm{K}$ for which this holdstrue.
Note that period of an idempotent matrix is 1 .

(c) Nilpotent Matrix : A square matrix is said to be nilpotent matrix of order $m, m \in N$, if $A^{m}=O, A^{m-1} \neq O$
Note that a nilpotent matrix will not be invertible.

(d) Involutary Matrix : If $A^{2}=I$, the matrix is said to be an involutary matrix.
Note that $A=A^{-1}$ for an involutary matrix.

(e) If $A$ and $B$ are square matrices of same order and $A B=B A$ then
$(\mathrm{A}+\mathrm{B})^{\mathrm{n}}={ }^{\mathrm{n}} \mathrm{C}_{0} \mathrm{~A}^{\mathrm{n}}+{ }^{\mathrm{n}} \mathrm{C}_{1} \mathrm{~A}^{\mathrm{n}-1} \mathrm{~B}+{ }^{\mathrm{n}} \mathrm{C}_{2} \mathrm{~A}^{\mathrm{n}-2} \mathrm{~B}^{2}$$+\ldots \ldots$$\ldots . .+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}} \mathrm{B}^{\mathrm{n}}$

---

---

11. SYMMETRIC & SKEW SYMMETRIC MATRIX :

(a) Symmetric matrix :

For symmetric matrix $\mathbf{A}=\mathbf{A}^{\mathbf{T}}$ i.e. $a_{i j}=a_{\mathrm{j}} \,\, \forall \,\, \mathrm{i}, \mathrm{j}$
Note : Maximum number of distinct entries in any symmetric matrix of order $\mathrm{n}$ is $\dfrac{\mathrm{n}(\mathrm{n}+1)}{2}$.

(b) Skew symmetric matrix :

Square matrix $\mathrm{A}=\left[a_{\mathrm{ij}}\right]$ is said to be skew symmetric if $\mathrm{A}^{\mathrm{T}}=-\mathrm{A}$ i.e. $\mathrm{a}_{\text {ii }}=-\mathrm{a}_{\mathrm{ii}} \,\, \forall \,\, \mathrm{i} \, \& \, \mathrm{j} .$ Hence if $\mathrm{A}$ is skew symmetric, then $\mathrm{a}_{\mathrm{ii}}=-\mathrm{a}_{\mathrm{ii}} \Rightarrow \mathrm{a}_{\mathrm{ii}}=0 \,\, \forall \,\, \mathrm{i}$
Thus the diagonal elements of a skew square matrix are all zero, but not the converse.

(c) Properties of symmetric & skew symmetric matrix :

(i) Let $A$ be any square matrix then, $A+A^{T}$ is a symmetric matrix & $A-A^{T}$ is a skew symmetric matrix.
(ii) The sum of two symmetric matrix is a symmetric matrix and the sum of two skew symmetric matrix is a skew symmetric matrix.

(iii) If $A$ & $B$ are symmetric matrices then,
(1) $\mathrm{AB}+\mathrm{BA}$ is a symmetric matrix
(2) $\mathrm{AB}-\mathrm{BA}$ is a skew symmetric matrix.

(iv) Every square matrix can be uniquely expressed as a sum or difference of a symmetric and a skew symmetric matrix.
$\begin{aligned}A &=\underbrace{\frac{1}{2}\left(\mathrm{~A}+\mathrm{A}^{\mathrm{T}}\right)}_{\text {symmetric }}+\underbrace{\frac{1}{2}\left(\mathrm{~A}-\mathrm{A}^{\mathrm{T}}\right)}_{\text {skew symmetric }} \\\text { and } \quad A &=\frac{1}{2}\left(\mathrm{~A}^{\mathrm{T}}+\mathrm{A}\right)-\frac{1}{2}\left(\mathrm{~A}^{\mathrm{T}}-\mathrm{A}\right)\end{aligned}$

---

---

12. ADJOINT OF A SQUARE MATRIX :

Let $A=\left[a_{i j}\right]=\left(\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right)$ be a square matrix and let the matrix formed by the cofactors of $\left[a_{i j}\right]$ in determinant $|A|$ is $\left(\begin{array}{lll}\mathrm{C}_{11} & \mathrm{C}_{12} & \mathrm{C}_{13} \\ \mathrm{C}_{21} & \mathrm{C}_{22} & \mathrm{C}_{23} \\ \mathrm{C}_{31} & \mathrm{C}_{32} & \mathrm{C}_{33}\end{array}\right) .$ Then $(\operatorname{adj} \mathrm{A})=\left(\begin{array}{ccc}\mathrm{C}_{11} & \mathrm{C}_{21} & \mathrm{C}_{31} \\ \mathrm{C}_{12} & \mathrm{C}_{22} & \mathrm{C}_{32} \\ \mathrm{C}_{13} & \mathrm{C}_{23} & \mathrm{C}_{33}\end{array}\right)$= Transpose of cofactor matrix. 

Note :
If $A$ be a square matrix of order $n$, then
(i) $\quad A(\operatorname{adj} A)=|A| I_{n}=(\operatorname{adj} A) \cdot A$
(ii) $\quad|\operatorname{adj} A|=|A|^{n-1}, n \geq 2$
(iii) $\operatorname{adj}(\operatorname{adj} A)=|A|^{n-2} A, \quad|A| \neq 0$.
(iv) $\operatorname{adj}(\mathrm{AB})=(\operatorname{adj} B)(\operatorname{adj} \mathrm{A})$
(v) $\operatorname{adj}(\mathrm{KA})=\mathrm{K}^{\mathrm{n}-1}(\operatorname{adj} \mathrm{A})$, where $\mathrm{K}$ is a scalar

---

---

13. INVERSE OF A MATRIX (Reciprocal Matrix) :

A square matrix A said to be invertible (non singular) if there exists a matrix $B$ such that, $A B=I \quad$ (Note that $A B=I \Leftrightarrow B A=I$ ) $B$ is called the inverse (reciprocal) of $A$ and is denoted by $A^{-1}$. Thus $\mathrm{A}^{-1}=\mathrm{B} \Leftrightarrow \mathrm{AB}=\mathrm{I}=\mathrm{BA}$
We have, $A \cdot(\operatorname{adj} A)=|A| I_{n}$
$\Rightarrow \mathrm{A}^{-1} \cdot \mathrm{A}(\operatorname{adj} \mathrm{A})=\mathrm{A}^{-1} \mathrm{I}_{\mathrm{n}}|\mathrm{A}|$
$\Rightarrow \mathrm{I}_{\mathrm{n}}(\operatorname{adj} \mathrm{A})=\mathrm{A}^{-1}|\mathrm{~A}| \mathrm{I}_{\mathrm{n}}$
$\therefore \quad \mathrm{A}^{-1}=\frac{(\operatorname{adj} \mathrm{A})}{|\mathrm{A}|}$

Note : The necessary and sufficient condition for a square matrix $\mathrm{A}$ to be invertible is that $|\mathrm{A}| \neq 0$

Theorem : If $A$ & $B$ are invertible matrices of the same order, then $(\mathrm{AB})^{-1}=\mathrm{B}^{-1} \mathrm{~A}^{-1}$

Note:
(i) If $A$ be an invertible matrix, then $A^{T}$ is also invertible & $\left(\mathrm{A}^{\mathrm{T}}\right)^{-1}=\left(\mathrm{A}^{-1}\right)^{\mathrm{T}}$

(ii) If $A$ is invertible,
(a) $\left(\mathrm{A}^{-1}\right)^{-1}=\mathrm{A}$
(b) $\left(A^{k}\right)^{-1}=\left(A^{-1}\right)^{k}=A^{-k} ; k \in N$

(iii) $\left|A^{-1}\right|=\dfrac{1}{|A|}$.

---

---

14. SYSTEM OF EQUATION & CRITERIA FOR CONSISTENCY Gauss - Jordan method:

Example :
$a_{1} x+b_{1} y+c_{1} z=d_{1}$
$a_{2} x+b_{2} y+c_{2} z=d_{2}$
$a_{3} x+b_{3} y+c_{3} z=d_{3}$

$\Rightarrow\left[\begin{array}{l}a_{1} x+b_{1} y+c_{1} z \\ a_{2} x+b_{2} y+c_{2} z \\ a_{3} x+b_{3} y+c_{3} z\end{array}\right]=\left[\begin{array}{l}d_{1} \\ d_{2} \\ d_{3}\end{array}\right]$$\Rightarrow\left[\begin{array}{lll}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3}\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}d_{1} \\ d_{2} \\ d_{3}\end{array}\right]$

$\Rightarrow A X=B \quad \Rightarrow A^{-1} A X=A^{-1} B$, if $|A| \neq 0$

$\Rightarrow X=A^{-1} B=\dfrac{\operatorname{Adj} A}{|A|} \cdot B$

Note:
(i) If | $A \mid \neq 0$, system is consistent having unique solution
(ii) If $|A| \neq 0$ & $(\operatorname{adj} A) \cdot B \neq O$ (Null matrix), system is consistent having unique non-trivial solution.
(iii) If $|A| \neq 0$ & $(\operatorname{adj} A) \cdot B=O$ (Null matrix), system is consistent having trivial solution. 
(iv) If $|A|=0$, then

---

---