Matrices
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:
-
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
-
$ \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} $ $|A| = 0$.