Matrices and Determinants

Definitions

$$\mathrm{An}\:{m}×{n}\:\mathrm{matrix}\:\boldsymbol{{A}}\:\mathrm{is}\:\mathrm{a}\:\mathrm{rectangular} \\ $$

$$\mathrm{array}\:\mathrm{of}\:\mathrm{elements}\:\mathrm{with}\:{m}\:\mathrm{rows}\:\mathrm{and}\:{n} \\ $$

$$\mathrm{colums} \\ $$

$$\boldsymbol{{A}}=\left[{a}_{{ij}} \right]=\begin{bmatrix}{{a}_{\mathrm{11}} }&{{a}_{\mathrm{12}} }&{\ldots}&{{a}_{\mathrm{1}{n}} }\\{{a}_{\mathrm{21}} }&{{a}_{{zz}} }&{\ldots}&{{a}_{\mathrm{2}{n}} }\\{\vdots}&{\vdots}&{}&{\vdots}\\{{a}_{{m}\mathrm{1}} }&{{a}_{{m}\mathrm{2}} }&{\ldots}&{{a}_{{mn}} }\end{bmatrix} \\ $$

Square Matrix

$$\mathrm{Square}\:\mathrm{matrix}\:\mathrm{is}\:\mathrm{of}\:\mathrm{order}\:{n}×{n}. \\ $$

$$\mathrm{A}\:\mathrm{square}\:\mathrm{matrix}\:\left[{a}_{{ij}} \right]\:\mathrm{is}\:\mathrm{symmtrc}\:\mathrm{if}\:{a}_{{ij}} ={a}_{{ji}.} \\ $$

$$\mathrm{A}\:\mathrm{square}\:\mathrm{matrix}\:\left[{a}_{{ij}} \right]\:\mathrm{is}\:\mathrm{skew}−\mathrm{symmtrc}\:\mathrm{if}\:{a}_{{ij}} =−{a}_{{ji}.} \\ $$

Diagonal Matrix

$$\mathrm{Diagonal}\:\mathrm{matrix}\:\mathrm{is}\:\mathrm{a}\:\mathrm{square}\:\mathrm{matrix}\:\mathrm{with}\:\mathrm{all} \\ $$

$$\mathrm{elements}\:\mathrm{zero}\:\mathrm{except}\:\mathrm{those}\:\mathrm{on}\:\mathrm{the}\:\mathrm{leading} \\ $$

$$\mathrm{diagonal}. \\ $$

Unit Matrix

$$\mathrm{Unit}\:\mathrm{matrix}\:\mathrm{is}\:\mathrm{a}\:\mathrm{diagonal}\:\mathrm{matrix}\:\mathrm{in}\:\mathrm{which} \\ $$

$$\mathrm{all}\:\mathrm{elements}\:\mathrm{on}\:\mathrm{the}\:\mathrm{leading}\:\mathrm{diagonal} \\ $$

$$\mathrm{are}\:\mathrm{1}.\:\mathrm{Unit}\:\mathrm{matrix}\:\mathrm{is}\:\mathrm{denoted}\:\mathrm{by}\:\boldsymbol{{I}}. \\ $$

Null Matrix

$$\mathrm{A}\:\mathrm{null}\:\mathrm{matrix}\:\mathrm{is}\:\mathrm{one}\:\mathrm{whose}\:\mathrm{all}\:\mathrm{elements}\:\mathrm{are}\:\mathrm{0}. \\ $$

Operations with Matrics

Addition/Subtraction

$$\mathrm{Two}\:\mathrm{matrices}\:\boldsymbol{{A}}\:\mathrm{and}\:\boldsymbol{{B}}\:\mathrm{are}\:\mathrm{equal}\:\mathrm{if}\:\mathrm{and} \\ $$

$$\mathrm{only}\:\mathrm{if}\:\mathrm{they}\:\mathrm{are}\:\mathrm{both}\:\mathrm{the}\:\mathrm{same}\:\mathrm{shape}\:\mathrm{and} \\ $$

$$\mathrm{corresponding}\:\mathrm{elements}\:\mathrm{are}\:\mathrm{equal}. \\ $$

$$\mathrm{Two}\:\mathrm{matrices}\:\mathrm{can}\:\mathrm{be}\:\mathrm{added}\:\left(\mathrm{or}\:\mathrm{subtracted}\right) \\ $$

$$\mathrm{if}\:\mathrm{and}\:\mathrm{only}\:\mathrm{if}\:\mathrm{they}\:\mathrm{have}\:\mathrm{the}\:\mathrm{same}\:\mathrm{shape}\:{m}×{n}. \\ $$

$$\boldsymbol{{A}}=\left[{a}_{{ij}} \right]=\begin{bmatrix}{{a}_{\mathrm{11}} }&{{a}_{\mathrm{12}} }&{\ldots}&{{a}_{\mathrm{1}{n}} }\\{{a}_{\mathrm{21}} }&{{a}_{\mathrm{22}} }&{\ldots}&{{a}_{\mathrm{2}{n}} }\\{\vdots}&{\vdots}&{}&{\vdots}\\{{a}_{{m}\mathrm{1}} }&{{a}_{{m}\mathrm{2}} }&{\ldots}&{{a}_{{mn}} }\end{bmatrix} \\ $$

$$\boldsymbol{{B}}=\left[{b}_{{ij}} \right]=\begin{bmatrix}{{b}_{\mathrm{11}} }&{{b}_{\mathrm{12}} }&{\ldots}&{{b}_{\mathrm{1}{n}} }\\{{b}_{\mathrm{21}} }&{{b}_{\mathrm{22}} }&{\ldots}&{{b}_{\mathrm{2}{n}} }\\{\vdots}&{\vdots}&{}&{\vdots}\\{{b}_{{m}\mathrm{1}} }&{{b}_{{m}\mathrm{2}} }&{\ldots}&{{b}_{{mn}} }\end{bmatrix} \\ $$

$$\boldsymbol{{A}}+\boldsymbol{{B}}=\left[{a}_{{ij}} +{b}_{{ij}} \right]=\begin{bmatrix}{{a}_{\mathrm{11}} +{b}_{\mathrm{11}} }&{{a}_{\mathrm{12}} +{b}_{\mathrm{12}} }&{\ldots}&{{a}_{\mathrm{1}{n}} +{b}_{\mathrm{1}{n}} }\\{{a}_{\mathrm{21}} +{b}_{\mathrm{21}} }&{{a}_{\mathrm{22}} +{b}_{\mathrm{22}} }&{\ldots}&{{a}_{\mathrm{2}{n}} +{b}_{\mathrm{2}{n}} }\\{\vdots}&{\vdots}&{}&{\vdots}\\{{a}_{{m}\mathrm{1}} +{b}_{{m}\mathrm{1}} }&{{a}_{{m}\mathrm{2}} +{b}_{{m}\mathrm{2}} }&{\ldots}&{{a}_{{mn}} +{b}_{{mn}} }\end{bmatrix} \\ $$

Scaler Multiplication

$$\mathrm{If}\:{k}\:\mathrm{is}\:\mathrm{a}\:\mathrm{scaler},\:\mathrm{and}\:\boldsymbol{{A}}\:\mathrm{is}\:\mathrm{a}\:\mathrm{matrix},\:\mathrm{then} \\ $$

$${k}\boldsymbol{{A}}=\left[{ka}_{{ij}} \right]=\begin{bmatrix}{{ka}_{\mathrm{11}} }&{{ka}_{\mathrm{12}} }&{\ldots}&{{ka}_{\mathrm{1}{n}} }\\{{ka}_{\mathrm{21}} }&{{ka}_{\mathrm{22}} }&{\ldots}&{{ka}_{\mathrm{2}{n}} }\\{\vdots}&{\vdots}&{}&{\vdots}\\{{ka}_{{m}\mathrm{1}} \:}&{{ka}_{{m}\mathrm{2}} }&{\ldots}&{{ka}_{{mn}} }\end{bmatrix} \\ $$

Multiplication

$$\mathrm{Two}\:\mathrm{matrices}\:\mathrm{can}\:\mathrm{be}\:\mathrm{multiplied}\:\mathrm{together} \\ $$

$$\mathrm{only}\:\mathrm{when}\:\mathrm{number}\:\mathrm{of}\:\mathrm{colums}\:\mathrm{in}\:\mathrm{the}\:\mathrm{first} \\ $$

$$\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{number}\:\mathrm{of}\:\mathrm{rows}\:\mathrm{in}\:\mathrm{the}\:\mathrm{second}. \\ $$

$$\mathrm{If} \\ $$

$$\boldsymbol{{A}}=\left[{a}_{{ij}} \right]=\begin{bmatrix}{{a}_{\mathrm{11}} }&{{a}_{\mathrm{12}} }&{\ldots}&{{a}_{\mathrm{1}{n}} }\\{{a}_{\mathrm{21}} }&{{a}_{\mathrm{22}} }&{\ldots}&{{a}_{\mathrm{2}{n}} }\\{\vdots}&{\vdots}&{}&{\vdots}\\{{a}_{{m}\mathrm{1}} \:}&{{a}_{{m}\mathrm{2}} }&{\ldots}&{{a}_{{mn}} }\end{bmatrix} \\ $$

$$\mathrm{and} \\ $$

$$\boldsymbol{{B}}=\left[{b}_{{ij}} \right]=\begin{bmatrix}{{b}_{\mathrm{11}} }&{{b}_{\mathrm{12}} }&{\ldots}&{{b}_{\mathrm{1}{k}} }\\{{b}_{\mathrm{21}} }&{{b}_{\mathrm{22}} }&{\ldots}&{{b}_{\mathrm{2}{k}} }\\{\vdots}&{\vdots}&{}&{\vdots}\\{{b}_{{n}\mathrm{1}} \:}&{{b}_{{n}\mathrm{2}} }&{\ldots}&{{b}_{{nk}} }\end{bmatrix} \\ $$

$$\boldsymbol{{C}}=\boldsymbol{{AB}}=\left[{c}_{{ij}} \right]\:\mathrm{where} \\ $$

$${c}_{{ij}} =\underset{\lambda=\mathrm{1}} {\overset{{n}} {\sum}}{a}_{{i}\lambda} {b}_{\lambda{j}} \\ $$

$$\mathrm{If}\:{m}×{n}\:\mathrm{matrix}\:\mathrm{is}\:\mathrm{multiplied}\:\mathrm{with}\:{n}×{k} \\ $$

$$\mathrm{matrix}\:\mathrm{then}\:\mathrm{result}\:\mathrm{is}\:\mathrm{a}\:{m}×{k}\:\mathrm{matrix}. \\ $$

Transpose of a Matrix

$$\mathrm{If}\:\mathrm{the}\:\mathrm{rows}\:\mathrm{and}\:\mathrm{columns}\:\mathrm{of}\:\mathrm{a}\:\mathrm{matrix}\:\mathrm{are} \\ $$

$$\mathrm{interchanged}\:\mathrm{then}\:\mathrm{the}\:\mathrm{new}\:\mathrm{matrix}\:\mathrm{is} \\ $$

$$\mathrm{called}\:\mathrm{the}\:\mathrm{transpose}\:\mathrm{of}\:\mathrm{the}\:\mathrm{original}\:\mathrm{matrix}. \\ $$

$$\mathrm{If}\:\boldsymbol{{A}}\:\mathrm{is}\:\mathrm{the}\:\mathrm{original}\:\mathrm{matrix},\:\mathrm{its}\:\mathrm{transpose} \\ $$

$$\mathrm{is}\:\mathrm{denoted}\:\boldsymbol{{A}}^{\mathrm{T}} . \\ $$

$$\mathrm{If}\:\boldsymbol{{AA}}^{\mathrm{T}} =\boldsymbol{{I}}\:\mathrm{then}\:\boldsymbol{{A}}\:\mathrm{is}\:\mathrm{orthogonal}\:\mathrm{matrix}. \\ $$

$$\mathrm{If}\:\boldsymbol{{AB}}\:\mathrm{is}\:\mathrm{defined},\:\mathrm{then} \\ $$

$$\left(\boldsymbol{{AB}}\right)^{\mathrm{T}} =\boldsymbol{{B}}^{\mathrm{T}} \boldsymbol{{A}}^{\mathrm{T}} \\ $$

Positive Integral Powers

$$\mathrm{If}\:{A}\:\mathrm{is}\:\mathrm{an}\:{n}×{n}\:\mathrm{matrix},\:\mathrm{then}\:\mathrm{we}\:\mathrm{define} \\ $$

$${A}^{\mathrm{2}} =\boldsymbol{{A}}{A},\:\boldsymbol{{A}}^{\mathrm{3}} =\left(\boldsymbol{{AAA}}\right)\:\mathrm{and}\:\mathrm{so}\:\mathrm{on}. \\ $$

$$\mathrm{In}\:\mathrm{general},\:\boldsymbol{{A}}^{{n}} =\left(\boldsymbol{{AA}}…{n}\:\mathrm{times}\right). \\ $$

$$\mathrm{Also},\:\:\:\boldsymbol{{A}}^{\mathrm{0}} =\boldsymbol{{I}}_{{n}} \:\mathrm{where}\:\boldsymbol{{I}}_{{n}} \:\mathrm{is}\:\mathrm{an}\:\mathrm{identity} \\ $$

$$\mathrm{matrix}\:\mathrm{of}\:\mathrm{order}\:{n}. \\ $$

$$ \\ $$

Matrix Polynomial

$$\mathrm{Let}\:{f}\left({x}\right)=\underset{{i}=\mathrm{0}} {\overset{\mathrm{m}} {\sum}}{a}_{{i}} {x}^{{i}} \:\mathrm{be}\:\mathrm{a}\:\mathrm{polynomial}\:\mathrm{of} \\ $$

$$\mathrm{degree}\:\mathrm{m},\:\mathrm{If}\:\boldsymbol{{A}}\:\mathrm{is}\:\mathrm{a}\:\mathrm{square}\:\mathrm{matrix}\:\mathrm{of}\: \\ $$

$$\mathrm{order}\:{n}.\:\mathrm{Then}\:\mathrm{we}\:\mathrm{define} \\ $$

$${f}\left(\boldsymbol{{A}}\right)=\underset{{i}=\mathrm{0}} {\overset{{m}} {\sum}}{a}_{{i}} \boldsymbol{{A}}^{{i}} \\ $$

Properties of Matrix Operations

$$\mathrm{1}.\:\mathrm{Matrix}\:\mathrm{addition}\:\mathrm{is}\:\mathrm{commutative}. \\ $$

$$\boldsymbol{{A}}+\boldsymbol{{B}}=\boldsymbol{{B}}+\boldsymbol{{A}} \\ $$

$$\mathrm{2}.\:\mathrm{Matrix}\:\mathrm{addition}\:\mathrm{is}\:\mathrm{associative}. \\ $$

$$\left(\boldsymbol{{A}}+\boldsymbol{{B}}\right)+\boldsymbol{{C}}=\boldsymbol{{A}}+\left(\boldsymbol{{B}}+\boldsymbol{{C}}\right) \\ $$

$$\mathrm{3}.\:\mathrm{If}\:\boldsymbol{{A}}\:\mathrm{is}\:\mathrm{an}\:{m}×{n}\:\mathrm{matrix}\:\mathrm{and}\:\boldsymbol{{O}}\:\mathrm{is}\:\mathrm{a}\:{m}×{n}\:\mathrm{null} \\ $$

$$\mathrm{matrix},\:\mathrm{then} \\ $$

$$\boldsymbol{{A}}+\boldsymbol{{O}}=\boldsymbol{{O}}+\boldsymbol{{A}}=\boldsymbol{{A}} \\ $$

$$\mathrm{4}.\:\mathrm{Negative}\:\mathrm{of}\:\mathrm{a}\:\mathrm{matrix}\:\boldsymbol{{A}}=\left[{a}_{{ij}} \right]_{{m}×{n}} \\ $$

$$−\boldsymbol{{A}}=\left[−{a}_{{ij}} \right]_{{m}×{n}} \\ $$

$$\mathrm{5}.\:\mathrm{Subtration}\:\mathrm{of}\:\mathrm{two}\:\mathrm{matrices} \\ $$

$$\boldsymbol{{A}}−\boldsymbol{{B}}=\boldsymbol{{A}}+\left(−\boldsymbol{{B}}\right) \\ $$

$$\mathrm{6}.\:\boldsymbol{{A}}−\boldsymbol{{A}}=\boldsymbol{{A}}+\left(−\boldsymbol{{A}}\right)=\left(−\boldsymbol{{A}}\right)+\boldsymbol{{A}}=\boldsymbol{{O}} \\ $$

$$\mathrm{7}.\:{k}\left(\boldsymbol{{A}}+\boldsymbol{{B}}\right)={k}\boldsymbol{{A}}+{k}\boldsymbol{{B}},\:\mathrm{where}\:{k}\:\mathrm{scaler} \\ $$

$$\mathrm{8}.\:\left({k}_{\mathrm{1}} +{k}_{\mathrm{2}} \right)\boldsymbol{{A}}={k}_{\mathrm{1}} \boldsymbol{{A}}+{k}_{\mathrm{2}} \boldsymbol{{A}},\:\mathrm{where}\:{k}_{\mathrm{1}} ,{k}_{\mathrm{2}} \:\mathrm{scaler} \\ $$

$$\mathrm{9}.\:{k}_{\mathrm{1}} \left({k}_{\mathrm{2}} \boldsymbol{{A}}\right)=\left({k}_{\mathrm{1}} {k}_{\mathrm{2}} \right)\boldsymbol{{A}},\:\mathrm{where}\:{k}_{\mathrm{1}} ,{k}_{\mathrm{2}} \:\mathrm{scaler} \\ $$

$$\mathrm{10}.\:\left(\boldsymbol{{A}}^{\mathrm{T}} \right)^{\mathrm{T}} =\boldsymbol{{A}} \\ $$

$$\mathrm{11}.\:\left(\boldsymbol{{A}}+\boldsymbol{{B}}\right)^{\mathrm{T}} =\boldsymbol{{A}}^{\mathrm{T}} +\boldsymbol{{B}}^{\mathrm{T}} \\ $$

$$\mathrm{12}.\:\left({k}\boldsymbol{{A}}\right)^{\mathrm{T}} ={k}\boldsymbol{{A}}^{\mathrm{T}} ,\:{k}\:\:\:\mathrm{scaler} \\ $$

$$\mathrm{13}.\:\mathrm{Symmetric}\:\mathrm{matrix}\:\:\:\boldsymbol{{A}}^{\mathrm{T}} =\boldsymbol{{A}} \\ $$

$$\mathrm{14}.\:\mathrm{Skew}−\mathrm{symmetric}\:\mathrm{matrix}\:\boldsymbol{{A}}^{\mathrm{T}} =−\boldsymbol{{A}} \\ $$

$$\mathrm{15}.\:\mathrm{Every}\:\mathrm{diagonal}\:\mathrm{element}\:\mathrm{of} \\ $$

$$\:\:\:\:\:\:\:\:\:\mathrm{skew}−\mathrm{symmetric}\:\mathrm{matrix}\:\mathrm{is}\:\mathrm{0}. \\ $$

$$\mathrm{16}.\:\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{two}\:\mathrm{symmetric}\:\mathrm{matrix}\:\mathrm{is}\:\mathrm{symmetric}. \\ $$

$$\mathrm{17}.\:\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{two}\:\mathrm{skew}−\mathrm{symmetric}\:\mathrm{matrix}\:\mathrm{is}\:\mathrm{skew}−\mathrm{symmetric}. \\ $$

$$\mathrm{18}.\:\mathrm{For}\:\mathrm{any}\:\mathrm{square}\:\mathrm{matrix} \\ $$

$$\:\:\:\:\:\:\:\:\:\left(\boldsymbol{{A}}+\boldsymbol{{A}}^{\mathrm{T}} \right)\:\mathrm{is}\:\mathrm{symmetric} \\ $$

$$\:\:\:\:\:\:\:\:\:\left(\boldsymbol{{A}}−\boldsymbol{{A}}^{\mathrm{T}} \right)\:\mathrm{is}\:\mathrm{skew}−\mathrm{symmetric} \\ $$

$$\mathrm{19}.\:\mathrm{Matrix}\:\mathrm{multiplication}\:\mathrm{is}\:\mathrm{not} \\ $$

$$\:\:\:\:\:\:\:\:\:\mathrm{commutative}\:\mathrm{is}\:\mathrm{general}. \\ $$

$$\mathrm{20}.\:\mathrm{Matrix}\:\mathrm{multiplication}\:\mathrm{is}\:\mathrm{associative}. \\ $$

$$\:\:\:\:\:\:\:\:\:\left(\boldsymbol{{AB}}\right)\boldsymbol{{C}}=\boldsymbol{{A}}\left(\boldsymbol{{BC}}\right) \\ $$

$$\mathrm{21}.\:\mathrm{Multiplication}\:\mathrm{distributes}\:\mathrm{addition}. \\ $$

$$\:\:\:\:\:\:\:\:\:\boldsymbol{{A}}\left(\boldsymbol{{B}}+\boldsymbol{{C}}\right)=\boldsymbol{{AB}}+\boldsymbol{{AC}} \\ $$

$$\:\:\:\:\:\:\:\:\:\left(\boldsymbol{{A}}+\boldsymbol{{B}}\right)\boldsymbol{{C}}=\boldsymbol{{AC}}+\boldsymbol{{BC}} \\ $$

$$\mathrm{22}.\:\mathrm{If}\:\boldsymbol{{A}}\:\mathrm{is}\:\mathrm{an}\:{m}×{n}\:\mathrm{matrix} \\ $$

$$\:\:\:\:\:\:\:\:\:\boldsymbol{{AI}}_{{n}} =\boldsymbol{{A}} \\ $$

$$\:\:\:\:\:\:\:\:\:\boldsymbol{{I}}_{{m}} \boldsymbol{{A}}=\boldsymbol{{A}} \\ $$

$$\:\:\:\:\:\:\:\:\:\mathrm{wherw}\:\boldsymbol{{I}}_{{n}} \:\mathrm{identity}\:\mathrm{matrix}\:\mathrm{of}\:\mathrm{order}\:{n}. \\ $$

$$\mathrm{23}.\:\mathrm{If}\:\boldsymbol{{A}}\:\mathrm{is}\:\mathrm{a}\:\mathrm{matrix}\:\mathrm{and}\:\boldsymbol{{O}}\:\mathrm{is}\:\mathrm{null}\:\mathrm{matrix}, \\ $$

$$\:\:\:\:\:\:\:\:\:\boldsymbol{{A}}_{{m}×{n}} \boldsymbol{{O}}_{{n}×{p}} =\boldsymbol{{O}}_{{m}×{p}} \\ $$

$$\:\:\:\:\:\:\:\:\:\boldsymbol{{O}}_{{p}×{m}} \boldsymbol{{A}}_{{m}×{n}} =\boldsymbol{{O}}_{{p}×{n}} \\ $$

$$\mathrm{24}.\:\mathrm{If}\:\boldsymbol{{A}}\:\mathrm{and}\:\boldsymbol{{B}}\:\mathrm{are}\:\mathrm{two}\:\mathrm{matrices}\:\mathrm{such}\:\mathrm{that} \\ $$

$$\:\:\:\:\:\:\:\:\:\boldsymbol{{AB}}\:\mathrm{is}\:\mathrm{defined}\:\mathrm{then} \\ $$

$$\:\:\:\:\:\:\:\:\:\left(\boldsymbol{{AB}}\right)^{\mathrm{T}} =\boldsymbol{{B}}^{\mathrm{T}} \boldsymbol{{A}}^{\mathrm{T}} \\ $$

$$\mathrm{25}.\:\mathrm{If}\:\boldsymbol{{A}}\:\mathrm{and}\:\boldsymbol{{B}}\:\mathrm{are}\:\mathrm{two}\:\mathrm{matrices}\:\mathrm{such}\:\mathrm{that} \\ $$

$$\:\:\:\:\:\:\:\:\:\boldsymbol{{AB}}\:\mathrm{is}\:\:\mathrm{defined}\:\:\mathrm{then} \\ $$

$$\:\:\:\:\:\:\:\:\:\:\boldsymbol{{A}}\left(−\boldsymbol{{B}}\right)=−\left(\boldsymbol{{AB}}\right) \\ $$

$$\:\:\:\:\:\:\:\:\:\:\left(−\boldsymbol{{A}}\right)\left(\boldsymbol{{B}}\right)=−\left(\boldsymbol{{AB}}\right) \\ $$

$$\mathrm{26}.\:\boldsymbol{{A}}\left(\boldsymbol{{B}}−\boldsymbol{{C}}\right)=\boldsymbol{{AB}}−\boldsymbol{{AC}} \\ $$

Determinants

$$\mathrm{Corresponding}\:\mathrm{to}\:\mathrm{each}\:\:\mathrm{square}\:\mathrm{matrix}\: \\ $$

$$\begin{bmatrix}{{a}_{\mathrm{11}} }&{\ldots}&{{a}_{\mathrm{1}{n}} }\\{\vdots}&{\ddots}&{\vdots}\\{{a}_{{n}\mathrm{1}} }&{\ldots}&{{a}_{{nn}} }\end{bmatrix} \\ $$

$$\boldsymbol{{A}}=\left[{a}_{{ij}} \right]\:\mathrm{there}\:\mathrm{is}\:\mathrm{associated}\:\mathrm{an}\:\mathrm{expression} \\ $$

$$\mathrm{called}\:\mathrm{the}\:{determinant}\:{of}\:\boldsymbol{{A}}\:\mathrm{denoted}\:\mathrm{by} \\ $$

$$\mathrm{det}\boldsymbol{{A}}\:\mathrm{or}\:\mid\boldsymbol{{A}}\mid,\:\mathrm{written}\:\mathrm{as} \\ $$

$$\mathrm{det}\boldsymbol{{A}}=\mid\boldsymbol{{A}}\mid=\begin{vmatrix}{{a}_{\mathrm{11}} }&{\ldots}&{{a}_{\mathrm{1}{n}} }\\{\vdots}&{\ddots}&{\vdots}\\{{a}_{{n}\mathrm{1}} }&{\ldots}&{{a}_{{nn}} }\end{vmatrix} \\ $$

Value of a Determinant

$$\mathrm{Value}\:\mathrm{of}\:\mathrm{determinant}\:\mathrm{of}\:\mathrm{order}\:\mathrm{2} \\ $$

$$\begin{vmatrix}{{a}_{\mathrm{11}} }&{{a}_{\mathrm{12}} }\\{{a}_{\mathrm{21}} }&{{a}_{\mathrm{22}} }\end{vmatrix}=\left({a}_{\mathrm{11}} {a}_{\mathrm{22}} −{a}_{\mathrm{12}} {a}_{\mathrm{21}} \right) \\ $$

$$\mathrm{Minor}\:\mathrm{of}\:{a}_{{ij}} \:\mathrm{in}\:\mid\boldsymbol{{A}}\mid \\ $$

$$\mathrm{The}\:\mathrm{minor}\:\mathrm{of}\:\mathrm{an}\:\mathrm{element}\:{a}_{{ij}} \:\mathrm{in}\:\mid\boldsymbol{{A}}\mid\:\mathrm{is}\:\mathrm{defined} \\ $$

$$\mathrm{as}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{determinant}\:\mathrm{obtained}\:\mathrm{by} \\ $$

$$\mathrm{deleting}\:{i}\mathrm{th}\:\mathrm{row}\:\mathrm{and}\:{j}\mathrm{th}\:\mathrm{column}\:\mathrm{of}\:\mid\boldsymbol{{A}}\mid, \\ $$

$$\mathrm{and}\:\mathrm{is}\:\mathrm{denoted}\:\mathrm{by}\:{M}_{{ij}} . \\ $$

$$\mathrm{Co}−\mathrm{factor}\:\mathrm{of}\:{a}_{{ij}} \:\mathrm{in}\:\mid\boldsymbol{{A}}\mid \\ $$

$$\mathrm{The}\:\mathrm{co}−\mathrm{factor}\:{C}_{{ij}} \:\mathrm{of}\:\mathrm{an}\:\mathrm{element}\:{a}_{{ij}} \:\mathrm{is} \\ $$

$$\mathrm{defined}\:\mathrm{as}\:{C}_{{ij}} =\left(−\mathrm{1}\right)^{{i}+{j}} \centerdot{M}_{{ij}} \\ $$

$$\mathrm{Value}\:\mathrm{of}\:\mathrm{a}\:\mathrm{determinant} \\ $$

$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\mathrm{a}\:\mathrm{determinant}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of} \\ $$

$$\mathrm{the}\:\mathrm{products}\:\mathrm{of}\:\mathrm{element}\:\mathrm{of}\:\mathrm{a}\:\mathrm{row}\:\left(\mathrm{or}\:\mathrm{column}\right) \\ $$

$$\mathrm{with}\:\mathrm{their}\:\mathrm{corresponding}\:\mathrm{co}−\mathrm{factors}. \\ $$

$$\mathrm{A}\:\mathrm{determinant}\:\:\mathrm{may}\:\mathrm{be}\:\mathrm{expanded}\:\mathrm{by} \\ $$

$$\mathrm{arbitarily}\:\mathrm{chosen}\:\mathrm{row}\:\mathrm{or}\:\mathrm{column}. \\ $$

$$\mathrm{Expansion}\:\mathrm{of}\:\mathrm{a}\:\mathrm{determinant}\:\mathrm{or}\:\mathrm{order}\:{n} \\ $$

$$\:\:\:\:\:\:\:\:\mathrm{Expansion}\:\mathrm{by}\:{i}\mathrm{th}\:\mathrm{row} \\ $$

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{det}\boldsymbol{{A}}=\underset{{j}=\mathrm{1}} {\overset{{n}} {\sum}}{a}_{{ij}} {C}_{{ij}} \\ $$

$$\:\:\:\:\:\:\:\:\mathrm{Expansion}\:\mathrm{by}\:{j}\mathrm{th}\:\mathrm{column} \\ $$

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{det}\boldsymbol{{A}}=\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}{a}_{{ij}} {C}_{{ij}} \\ $$

Properties of Determinants

$$\mathrm{1}.\:\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\mathrm{a}\:\mathrm{determinant}\:\mathrm{remains} \\ $$

$$\:\:\:\:\:\:\mathrm{unchanged}\:\mathrm{if}\:\mathrm{its}\:\mathrm{row}\:\mathrm{and}\:\mathrm{columns}\:\mathrm{are} \\ $$

$$\:\:\:\:\:\:\:\mathrm{interchanged}. \\ $$

$$\mathrm{2}.\:\:\mathrm{If}\:\mathrm{two}\:\mathrm{rows}\:\mathrm{or}\:\mathrm{columns}\:\mathrm{of}\:\mathrm{a}\:\mathrm{determinant}\:\mathrm{are}\: \\ $$

$$\:\:\:\:\:\:\:\mathrm{interchanged},\:\mathrm{the}\:\mathrm{sign}\:\mathrm{of}\:\mathrm{determinant} \\ $$

$$\:\:\:\:\:\:\:\mathrm{is}\:\mathrm{changed}\:\mathrm{but}\:\mathrm{the}\:\mathrm{absolute}\:\mathrm{value}\: \\ $$

$$\:\:\:\:\:\:\:\:\mathrm{remains}\:\mathrm{same}. \\ $$

$$\mathrm{3}.\:\:\:\mathrm{If}\:\mathrm{two}\:\mathrm{rows}\:\left(\mathrm{or}\:\mathrm{two}\:\mathrm{columns}\right)\:\mathrm{are} \\ $$

$$\:\:\:\:\:\:\:\:\mathrm{identical},\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{determinant} \\ $$

$$\:\:\:\:\:\:\:\:\mathrm{is}\:\mathrm{0}. \\ $$

$$\mathrm{4}.\:\:\:\mathrm{If}\:\mathrm{the}\:\mathrm{element}\:\mathrm{of}\:\mathrm{any}\:\mathrm{row}\:\mathrm{or}\:\mathrm{column} \\ $$

$$\:\:\:\:\:\:\:\:\mathrm{are}\:\mathrm{multiplied}\:\mathrm{by}\:\mathrm{a}\:\mathrm{common}\:\mathrm{factor}, \\ $$

$$\:\:\:\:\:\:\:\:\mathrm{the}\:\mathrm{determinant}\:\mathrm{is}\:\mathrm{multiplied}\:\mathrm{by} \\ $$

$$\:\:\:\:\:\:\:\:\mathrm{that}\:\mathrm{factor}. \\ $$

$$\mathrm{5}.\:\:\:\mathrm{If}\:\mathrm{the}\:\mathrm{elements}\:\mathrm{of}\:\mathrm{any}\:\mathrm{row}\:\left(\mathrm{or}\:\mathrm{column}\right) \\ $$

$$\:\:\:\:\:\:\:\mathrm{are}\:\mathrm{increased}\:\left(\mathrm{or}\:\mathrm{decreased}\right)\:\mathrm{by}\:\mathrm{equal} \\ $$

$$\:\:\:\:\:\:\:\mathrm{multiples}\:\mathrm{or}\:\mathrm{corresponding}\:\mathrm{elements}\:\mathrm{of} \\ $$

$$\:\:\:\:\:\:\:\mathrm{any}\:\mathrm{other}\:\mathrm{row}\:\left(\mathrm{or}\:\mathrm{column}\right),\:\mathrm{the}\:\mathrm{value} \\ $$

$$\:\:\:\:\:\:\:\mathrm{of}\:\mathrm{the}\:\mathrm{determinant}\:\mathrm{is}\:\mathrm{unchanged}. \\ $$

Adjoint of a Matrix

$$\mathrm{If}\:\boldsymbol{{A}}\:\mathrm{is}\:\mathrm{a}\:\mathrm{square}\:{n}×{n}\:\mathrm{matrix},\:\mathrm{its}\:\mathrm{adjoint}, \\ $$

$$\mathrm{denoted}\:\mathrm{by}\:\mathrm{adj}\boldsymbol{{A}},\:\mathrm{is}\:\mathrm{the}\:\mathrm{transpose}\:\mathrm{of}\:\mathrm{the} \\ $$

$$\mathrm{matrix}\:\mathrm{of}\:\mathrm{the}\:\mathrm{matrix}\:\mathrm{if}\:\mathrm{cofactors}\:{C}_{{ij}} \:\mathrm{of}\:\boldsymbol{{A}}. \\ $$

$$\mathrm{adj}\boldsymbol{{A}}=\left[{C}_{{ij}} \right]^{\mathrm{T}} \\ $$

Inverse of a Matrix

$$\mathrm{If}\:\boldsymbol{{A}}\:\mathrm{is}\:\mathrm{a}\:\mathrm{square}\:\mathrm{matrix}\:\mathrm{with}\:\mathrm{a}\:\mathrm{nonsingular} \\ $$

$$\mid\boldsymbol{{A}}\mid\:\mathrm{then}\:\mathrm{its}\:\mathrm{inverse}\:\boldsymbol{{A}}^{−\mathrm{1}} \:\mathrm{is}\:\mathrm{given}\:\mathrm{by}: \\ $$

$$\boldsymbol{{A}}^{−\mathrm{1}} =\:\frac{\mathrm{adj}\boldsymbol{{A}}}{\mathrm{det}\boldsymbol{{A}}} \\ $$

$$\mathrm{If}\:\mathrm{matrix}\:\mathrm{product}\:\boldsymbol{{AB}}\:\mathrm{is}\:\mathrm{defined}\:\mathrm{then} \\ $$

$$\left(\boldsymbol{{AB}}\right)^{−\mathrm{1}} =\boldsymbol{{B}}^{−\mathrm{1}} \boldsymbol{{A}}^{−\mathrm{1}} \\ $$

Invertible Matrix

$$\mathrm{1}.\:\boldsymbol{{AA}}^{−\mathrm{1}} =\boldsymbol{{A}}^{−\mathrm{1}} \boldsymbol{{A}}=\boldsymbol{{I}} \\ $$

$$\mathrm{2}.\:\boldsymbol{{AB}}=\boldsymbol{{AC}}\Rightarrow\boldsymbol{{B}}=\boldsymbol{{C}},\:\mathrm{if}\:\mid\boldsymbol{{A}}\mid\neq\mathrm{0} \\ $$

$$\mathrm{3}.\:\left(\boldsymbol{{AB}}\right)^{−\mathrm{1}} =\boldsymbol{{B}}^{−\mathrm{1}} \boldsymbol{{A}}^{−\mathrm{1}} ,\:\mathrm{if}\:\mid\boldsymbol{{A}}\mid,\mid\boldsymbol{{B}}\mid\neq\mathrm{0} \\ $$

$$\mathrm{4}.\:\left(\boldsymbol{{A}}^{\mathrm{T}} \right)^{−\mathrm{1}} =\left(\boldsymbol{{A}}^{−\mathrm{1}} \right)^{\mathrm{T}} ,\:\mathrm{if}\:\mid\boldsymbol{{A}}\mid\neq\mathrm{0},\:\mathrm{if}\:\mid\boldsymbol{{A}}\mid\neq\mathrm{0} \\ $$

$$\mathrm{5}.\:\mathrm{adj}\left(\boldsymbol{{AB}}\right)=\left(\mathrm{adj}\boldsymbol{{B}}\right)\left(\mathrm{adj}\boldsymbol{{A}}\right),\:\mathrm{if}\:\mid\boldsymbol{{A}}\mid,\mid\boldsymbol{{B}}\mid\neq\mathrm{0} \\ $$

$$\mathrm{6}.\:\left(\mathrm{adj}\boldsymbol{{A}}\right)^{\mathrm{T}} =\mathrm{adj}\boldsymbol{{A}}^{\mathrm{T}} ,\:\mathrm{if}\:\mid\boldsymbol{{A}}\mid\neq\mathrm{0} \\ $$

$$\mathrm{7}.\:\mid\mathrm{adj}\boldsymbol{{A}}\mid=\mid\boldsymbol{{A}}\mid^{{n}−\mathrm{1}} ,\:\mathrm{if}\:\mid\boldsymbol{{A}}\mid\neq\mathrm{0} \\ $$

$$\mathrm{8}.\:\mathrm{adj}\left(\mathrm{adj}\boldsymbol{{A}}\right)=\mid\boldsymbol{{A}}\mid^{{n}−\mathrm{2}} \boldsymbol{{A}} \\ $$

Elementary Row and Column Transformation of a Matrix

The following are three elementary transformation of a matrix $R_n$ indicates $nth$ row.

Interchange of any Two Rows or Two Columns ($R_m = R_n, R_n = R_m$ or $C_m = C_n, C_n = C_m$)
Multiplication of Row or Column by a Non-zero Number ($R_n=kR_n$ or $C_n=kC_n$ )
Multiplication of Row or Column by a Non-zero Number and Add the Result to the Other Row or Column ($R_n=R_n+kR_m$ or $C_n=C_n+kC_m$)

Linear Independence

Row $R_1, R_2, …, R_n$ with same number of columns are linearly independent if

$a_1R_1+a_2R_2+…+a_nR_n=R_0$, implies that $a_i=0$ for $i=1,2,…n$, $R_0$ is row with all columns with values 0.

Columns $C_1, C_2, …, C_n$ with same number of rows are linearly independent if

$a_1C_1+a_2C_2+…+a_nC_n=C_0$, implies that $a_i=0$ for $i=1,2,…n$, $C_0$ is column with all row with values 0.

Vectors $v_1, v_2, …, v_n$ with same number of dimensions are linearly independent if

$a_1v_1+a_2v_2+…+a_nv_n=0$, implies that $a_i=0$ for $i=1,2,…n$

Rank of a Matrix

The maximum number of linearly independent columns (or rows) of a matrix is called the rank of a matrix. The rank of a matrix cannot exceed the number of its rows or columns.

Key Concepts:

Linearly Independent Rows/Columns: A set of vectors (rows or columns) are linearly independent if no vector in the set can be written as a linear combination of the others. The rank indicates how many of the rows or columns are linearly independent.
Row Rank and Column Rank: The row rank is the number of linearly independent rows, and the column rank is the number of linearly independent columns. Row rank is always equal to column rank for any matrix, which is why we just call it the rank of the matrix.

Methods to Find the Rank:

Row Echelon Form (REF):
- Transform the matrix into row echelon form (REF) using Gaussian elimination.
- The rank is the number of non-zero rows in the REF.
Reduced Row Echelon Form (RREF):
- Transform the matrix into reduced row echelon form (RREF), which is a further simplified version of REF.
- The rank is the number of non-zero rows in the RREF.
Determinant Method (for square matrices):
- For a square matrix, if the determinant is non-zero, the rank is equal to the number of rows (or columns). If the determinant is zero, the matrix has rank less than its size.
Singular Value Decomposition (SVD):
- For a matrix AAA, the rank can also be determined by the number of non-zero singular values in its Singular Value Decomposition (SVD).

Minor Method: If the rank of matrix A is r, then there exists at least one minor of order r which does not vanish. Every minor of matrix A of order (r + 1) and higher-order (if any) vanishes.

Row Echelon Form

Specifically, a matrix is in row echelon form if

All rows consisting of only zeroes are at the bottom
The leading entry (that is the left-most nonzero entry) of every nonzero row is to the right of the leading entry of every row above
To convert a matrix to Row Echelon Form (REF), you perform a series of Gaussian elimination steps. The goal is to make the matrix satisfy the following conditions:
All nonzero rows are above any rows of all zeros.
The leading entry (also called the pivot) in each nonzero row is 1.
The pivot in any row appears to the right of the pivot in the row above it.
All entries below a pivot are zero.

Step-by-Step Process to Covert to Row Echelon From

Identify the first non-zero element in the first column. This element becomes your pivot.
Swap rows (if necessary) to make sure the pivot is at the top of the column (i.e., the pivot should be the first non-zero entry in the row).
Scale the row (if necessary) to make the pivot equal to 1. This can be done by dividing the entire row by the value of the pivot element.
Eliminate entries below the pivot: Use row operations to create zeros below the pivot. This is done by subtracting multiples of the pivot row from the rows below it.
Move to the next column: Once the pivot in the first column is dealt with, move to the next column and repeat the process for the remaining submatrix (i.e., ignore the row and column where the pivot was placed).
Continue until all pivots are in place: Repeat the process until you have processed all columns.

Reduced Row Echelon Form

To convert a matrix to Reduced Row Echelon Form (RREF), you need to apply Gaussian-Jordan elimination. The goal is to satisfy the following conditions:

The first non-zero entry in each row (the pivot) is 1.
The pivot is the only non-zero entry in its column (i.e., all entries above and below the pivot are zero).
The pivots in each row move to the right as you move down the rows.
All rows of zeros are at the bottom of the matrix.

Problems in Matrices and Determinants

https://www.tinkutara.com/category/maths/matricesanddeterminants/