\(A^{-1}=\left(\begin{array}{cc} 1 & -1 \\ 2 & 3 \end{array}\right)\)
\(A^2=\left(\begin{array}{cc} 1 & -1 \\ 2 & 3 \end{array}\right)\left(\begin{array}{cc} 1 & -1 \\ 2 & 3 \end{array}\right)\)
\(=\left(\begin{array}{ll} 1-2 & -1-3 \\ 2+6 & -2+9 \end{array}\right)=\left(\begin{array}{cc} -1 & -4 \\ 8 & 7 \end{array}\right)\)
\(\mathrm{LHS}=A^2-4 A+5 l_2\)
\(=\left(\begin{array}{cc} -1 & -4 \\ 8 & 7 \end{array}\right)-4\left(\begin{array}{cc} 1 & -1 \\ 2 & 3 \end{array}\right)+5\left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right)\)
\(=\left(\begin{array}{cc} -1 & -4 \\ 8 & 7 \end{array}\right)-\left(\begin{array}{cc} 4 & -4 \\ 8 & 12 \end{array}\right)+\left(\begin{array}{ll} 5 & 0 \\ 0 & 5 \end{array}\right)\)
\(=\left(\begin{array}{cc} -1 & -4 \\ 8 & 7 \end{array}\right)+\left(\begin{array}{cc} 1 & 4 \\ -8 & -7 \end{array}\right)=\left(\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right)\)
\(=O=R H S\)
\(\mathrm{A}^{-1}=\frac{1}{|A|} \operatorname{adj} A\)
\(|A|=3+2=5 \neq 0\)
\(A^{-1}=\frac{1}{5}\left(\begin{array}{cc} 3 & 1 \\ -2 & 1 \end{array}\right)\)
\(\frac { 1 }{ 5 } \left[ \begin{matrix} 3 & 1 \\ -2 & 1 \end{matrix} \right] \)

\(X=\left(\begin{array}{ccc} 8 & -1 & -3 \\ -5 & 1 & 2 \\ 10 & -1 & -4 \end{array}\right)\)
\(|X|=8(-4+2)+1(20-20)-3(5-10)\)
\(=-16+15=-1 \neq 0\)
\(A_{11}=8=-4+2=-2 \)
\(A_{12}=-1=-(20-20)=0\)
\(A_{13}=-3=5-10=-5\)
\(\mathrm{A}_{21}=-5=-(4-3)=-1 \)
\(\mathrm{A}_{22}= 1=-32+30=-2 \)
\(\mathrm{A}_{23}=2=-(-8+10)=-2\)
\(\mathrm{A}_{31}= 10=-2+3=1 \)
\(\mathrm{A}_{32}=-1=-(16-15)=-1 \)
\(\mathrm{A}_{33}=-4=8-5=+3\)
\(\text {Co-factor matrix }=\left(\begin{array}{ccc} -2 & 0 & -5 \\ -1 & -2 & -2 \\ 1 & -1 & +3 \end{array}\right)\)
\(\operatorname{Adj} X=\left(\begin{array}{ccc} -2 & -1 & 1 \\ 0 & -2 & -1 \\ -5 & -2 & +3 \end{array}\right)\)
\(X^{-1}=\frac{1}{|X|} \operatorname{adj} X\)
\(=\frac{-1}{1}\left(\begin{array}{ccc} -2 & -1 & 1 \\ 0 & -2 & -1 \\ -5 & -2 & +3 \end{array}\right)\)
\( Y=X^{-1}\)
\(\left(\begin{array}{ccc} 2 & 1 & -1 \\ 0 & 2 & 1 \\ 5 & p & q \end{array}\right)=\left(\begin{array}{ccc} 2 & 1 & -1 \\ 0 & 2 & 1 \\ 5 & 2 & -3 \end{array}\right)\)
p=2,q=-3

A=\(\left[ \begin{matrix} 1 & 1 & 3 \\ 2 & \lambda & 4 \\ 9 & 7 & 11 \end{matrix} \right] \)
|A| = 0
⇒ 11λ - 28 - 1(22 - 36) + 3(14 - 9λ) = 0
⇒ 11λ - 28 + 14 + 42 - 27λ = 0
⇒ -16λ + 28 = 0
⇒ 16λ = 28
⇒  λ = \(\frac{28}{16}=\frac{14}{8}=\frac{7}{4}\)
\(\lambda =\frac { 7 }{ 4 } \)

A=\(\left[ \begin{matrix} 2 & -2 & 2 \\ 2 & 3 & 0 \\ 9 & 1 & 5 \end{matrix} \right] \)
\(|A|=2(15-0)+2(10-0)+2(2-27)\)
\(=30+20-50=0\)
\((\operatorname{adj} \mathrm{A}) \mathrm{A}=|A| I=0\)

\(A B=\left(\begin{array}{cc} 2 & 3 \\ 1 & -6 \end{array}\right)\left(\begin{array}{cc} -1 & 4 \\ 1 & -2 \end{array}\right)\)
\(=\left(\begin{array}{cc} -2+3 & 8-6 \\ -1-6 & 4+12 \end{array}\right)=\left(\begin{array}{cc} 1 & 2 \\ -7 & 16 \end{array}\right)\)
\(\mathrm{LHS}=\operatorname{adj}(\mathrm{AB})=\left(\begin{array}{cc} 16 & -2 \\ 7 & 1 \end{array}\right)\)
\(\mathrm{RHS}=(\operatorname{adj} B)(\operatorname{adj} \mathrm{A})\)
\(=\left(\begin{array}{cc} -2 & -1 \\ -1 & -1 \end{array}\right)\left(\begin{array}{cc} -6 & -3 \\ -1 & 2 \end{array}\right)\)
\(=\left(\begin{array}{cc} 12+4 & 6-5 \\ 6+1 & 3-2 \end{array}\right)=\left(\begin{array}{cc} 16 & -2 \\ 7 & 1 \end{array}\right)\)
\(a d j(A B)=(a d j\ B)(a d j\ A)\)

Given A = \(\begin{bmatrix}1 &3&3 \\1 &4&3\\1&3&4 \end{bmatrix} \)
\(A_{11}=1=16-9=7\)
\(A_{12}=3=-(4-3)=-1\)
\(A_{13}= 3=3-4=-1\)
\(A_{21}=1=-(12-9)=-3 \)
\(A_{22}= 4=4-3=1 \)
\(A_{23}= 3=-(3-3)=0 \)
\( A_{31}= 1=9-12=-3 \)
\(A_{32}= 3=3-3=0 \)
\(A_{33}= 4=4-3=1\)
\(\text {Co-factor matrix }=\left(\begin{array}{ccc} 7 & -1 & -1 \\ -3 & 1 & 0 \\ -3 & 0 & 1 \end{array}\right)\)
A^-1 = \(\begin{bmatrix} 7&-3&-3\\-1&1&0\\-1&0&1 \end{bmatrix}\)
\(|A| =1(16-9)-3(4-3)+3(3-4) \)
\(=7-3-3=1 \neq 0\)
\(\therefore \mathrm{A}^{-1} \text { exists }\)
\(\mathrm{A}(\operatorname{adj} \mathrm{A})=\left(\begin{array}{lll} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{array}\right)\left(\begin{array}{ccc} 7 & -3 & -1 \\ -1 & 1 & 0 \\ -1 & 0 & 1 \end{array}\right)\)
\(=\left(\begin{array}{lll} 7-3-3 & -3+3+0 & -3+0+3 \\ 7-4-3 & -3+4+0 & -3+0+3 \\ 7-3-4 & -3+3+0 & -3+0+4 \end{array}\right)\)
\(=\left(\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right)=|A| I\)
\(=\mathrm{A}(\operatorname{adj} \mathrm{A})=|A| I\)
\(\mathrm{A}^{-1}=\frac{1}{|A|} \operatorname{adj} A=\left(\begin{array}{ccc} 7 & -3 & -3 \\ -1 & 1 & 0 \\ -1 & 0 & 1 \end{array}\right)\)

LHS = \(\begin{vmatrix}-a^2&ab&ac\\ab&-b^2&bc\\ac&bc&-c^2 \end{vmatrix}\)
R1 ஐ a ஆல், R2 ஐ b ஆல், R3 ஐ முறையே பெருக்கி, abc ஆல் வகுக்கிறது,
LHS = \(abc\begin{vmatrix}-a&b&c\\a&-b&c\\a&b&-c \end{vmatrix}\)
LHS = \(a^2b^2c^2\begin{vmatrix}-1&1&1\\1&-1&1\\1&1&-1 \end{vmatrix}\)
\(R_2 \rightarrow R_2+R_1, R_3 \rightarrow R_3+R_1\)
\(=a^2b^2c^2\begin{vmatrix}0&0&2\\1&-1&1\\1&1&-1 \end{vmatrix}\)
\(=a^2 b^2 c^2[-1(0-4)]=4 a^2 b^2 c^2\)
= RHS.

LHS = \(\begin{vmatrix}0 &ab^2 &ac^2 \\a^2b & 0 & bc^2\\a^2c&b^2c&0\end{vmatrix}\)
R₁, R₂, R₃
\(=\operatorname{abc}\left|\begin{array}{ccc} 0 & b^2 & c^2 \\ a^2 & 0 & c^2 \\ a^2 & b^2 & 0 \end{array}\right|\)
C₁, C₂, C₃
= \(a^2b^2c^2\begin{vmatrix} 0 & 1 & 1 \\ 1 & 0 & 1\\ 1 & 1 & 0 \end{vmatrix}\)
= a³b³c³ [0 - 1 (0 - 1)+ 1(1 - 0)]
= a³b³c³ (1 + 1) = 2 a³b³c³ = RHS

\(Adj \ A=​\left[ \begin{matrix} 4 & -3 \\ -1 & 2 \end{matrix} \right] \)

\(A=\left[ \begin{matrix} 2 & 4 \\ -3 & 2 \end{matrix} \right] \)
\(|A|=\left[ \begin{matrix} 2 & 4 \\ -3 & 2 \end{matrix} \right] \)
=16 ≠ 0
A என்பது பூச்சியமற்ற கோவை அணி ஆகையால்,A^-1யைக் காணஇயலும்.
adj \(A=\left[ \begin{matrix} 2 & -4 \\3 & 2 \end{matrix} \right] \)
\({ A }^{ -1 }=\frac { 1 }{ \left| A \right| } adjA\)
\(=\frac { 1 }{ 16 } \left[ \begin{matrix} 2 & -4 \\ 3 & 2 \end{matrix} \right] \)