Properties of Matrix Addition:

PYQ-2023-Matrices-Q1, PYQ-2023-Matrices-Q10

$\quad$ If $ A, B \ \text{and} \ C $ are matrices of same order, then

• (Commutative law) $$ A + B = B + A $$

• (Associative law) $$(A + B) + C = A + (B + C) $$

• (Existence of Additive Identity) $$ A + O = O + A = A$$

  where $O$ is zero matrix which is additive identity of the matrix

• (Additive Inverse) $$ A + B = O = B + A $$

  $B $ is called additive inverse of $A$ and also $A$ is called the additive inverse of $A$

Properties of Scalar Multiplication:

PYQ-2023-Matrices-Q7, PYQ-2023-Matrices-Q10, PYQ-2023-Matrices-Q11

$\quad $ If $ A, B \ \text{and} \ C $ are matrices of same order and $λ, µ $ are any two scalars then

• $ \quad \lambda (A + B) = \lambda A + \lambda B $

• $\quad (\lambda + \mu)A = \lambda A + \mu A$

• $\quad \lambda(\mu A) = (\lambda \mu)A = \mu(\lambda A)$

• $\quad (-\lambda A) = - (\lambda A) = \lambda(-A)$

• $\quad \text{tr} (kA) = k \text{tr} (A)$

Types of matrix:

PYQ-2023-Matrices-Q6

• Symmetric Matrix

  A square matrix $A=\left[a_{ij}\right]$ is called a symmetric matrix i $a_{ij}=a_{ji}$ for all $i, j$.

• Skew-Symmetric Matrix

  A square matrix $A$ is skew-symmetric when $a_{ij}=-a_{ji}$ for all $i, j$.

• Hermitian

  A square matrix $A=\left[a_{ij}\right]$ is called a Hermitian matrix if $A=A^\dagger$

• Skew-Hermitian

  A square matrix $A=\left[a_{ij}\right]$ is called a Skew-Hermitian matrix if $A=-A^\dagger$

• Orthogonal Matrix

  A square matrix $A$ is orthogonal if $AA^\top=I_n=A^\top A$.

• Idempotent Matrix

  A square matrix $A$ is idempotent if $A^2=A$.

• Involuntary Matrix Involuntary Matrix

  A square matrix $A$ is involuntary if $A^2=I$ or $A^{-1}=A$.

• Nilpotent Matrix

  A square matrix $A$ is nilpotent if there exists $p \in \mathbb{N} \ \text{such that} \ A^{p} = O$

Properties Of Trace Of Matrix

PYQ-2023-Matrices-Q12

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

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

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

Properties Of Transpose Of Matrix:

PYQ-2023-Matrices-Q5, PYQ-2023-Matrices-Q6, PYQ-2023-Matrices-Q11

• $ \left(A^T\right)^T=A$

• $ (A \pm B)^{\top}=A^{\top} \pm B^{\top}$

• $ (A B)^{\top}=B^{\top} A^{\top}$

• $ (k A)^{\top}=k(A)^{\top}$

• $ \left(A_1 A_2 A_3\right.$ ..$\left.A_{n-1} A_n\right)^{\top}=A_n^{\top} A_{n-1}^{\top}$ $A_3^{\top} A_2^{\top} A_1^{\top}$

• $ I^{T}=I$

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

Properties of matrix multiplication:

PYQ-2023-Matrices-Q1, PYQ-2023-Matrices-Q4, PYQ-2023-Matrices-Q5, PYQ-2023-Matrices-Q7, PYQ-2023-Matrices-Q9, PYQ-2023-Matrices-Q12, PYQ-2023-Matrices-Q15, PYQ-2023-Area_Under_Curves-Q14

• $ \mathrm{AB} \neq \mathrm{BA}$

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

• $ A \cdot(B+C)=A \cdot B+A \cdot C$

• $ \text{The product of two matrices can be a null matrix while neither of them is null, i.e. if} \ AB = 0 $,

  $\text{it is not necessary that either} \ $ A = O $ \text{or} $ B = O

• $ tr(AB) = tr(BA)$

• $ If $AB = AC ⇒ B ≠ C \ \text{(Cancellation Law is not applicable)} $

Properties of Adjoint of a Matrix:

PYQ-2023-Matrices-Q2, PYQ-2023-Matrices-Q3, PYQ-2023-Matrices-Q4, PYQ-2023-Matrices-Q10, PYQ-2023-Matrices-Q11, PYQ-2023-Matrices-Q14

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

• $|\operatorname{adj} A|=|A|^{n-1}$

• $(\operatorname{adj} A B)=(\operatorname{adj} B)(\operatorname{adj} A)$

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

• $(\operatorname{adj} K A)=K^{n-1}(\operatorname{adj} A)$

Properties of Inverse of a matrix:

PYQ-2023-Matrices-Q2, PYQ-2023-Matrices-Q5,PYQ-2023-Matrices-Q8, PYQ-2023-Matrices-Q14

$\quad \mathrm{A}^{-1}$ exists if $\mathrm{A}$ is non singular i.e. $|\mathrm{A}| \neq 0$

• $\mathrm{A}^{-1}=\frac{1}{|\mathrm{~A}|}(\operatorname{Adj} . \mathrm{A})$

• $\mathrm{A}^{-1} \mathrm{~A}=\mathrm{I}_{\mathrm{n}}=\mathrm{AA}^{-1}$

• $\left(A^{\top}\right)^{-1}=\left(A^{-1}\right)^{\top}$

• $\left(\mathrm{A}^{-1}\right)^{-1}=\mathrm{A}$

• $\left|A^{-1}\right|=|A|^{-1}=\frac{1}{|A|}$

Properties Of Positive Integral Powers Of A Square Matrix:

PYQ-2023-Matrices-Q5, PYQ-2023-Matrices-Q6, PYQ-2023-Matrices-Q7, PYQ-2023-Matrices-Q12, PYQ-2023-Matrices-Q15

• $ A^m A^n = A^{m+n} $

• $ (A^m)^n = A^{mn} = (A^n)^m$

• $ I^n = I$

• $ A^0 = I_n$

Solution to a System of Equations:

• If |$A$| ≠ $O$, $\text{then the system is consistent and has a unique solution, given by } X = A^{–1} B$

• If |$A$| =$ O$, and $(Adj A) B ≠ O$, $\text{then the system is inconsistent} $

• If |$A$| = $O$, and $(Adj A) B = O$, $\text{ then the system is consistent and has infinitely many solutions}$.

• $AX = O$ $\text{is known as a homogeneous system of linear equations, here }$ $B = 0$.

• $\text{A system of homogeneous equations is always consistent}$.

• $\text{The system has a non-trivial solution (non-zero solution), if} $ $|A| = 0$.