base

Trigonometry Identities

${\tan }^{2}x+1={\sec }^{2}x$ ${\cot }^{2}x+1={\csc }^{2}x$ $\sin (A+B)\times \sin (A-B)={\sin }^2(A)-{\sin }^2(B)$ $\cos (A+B)\times \cos (A-B)={\cos }^2(A)-{\sin }^2(B)={\cos }^2(B)-{\sin }^2(A)$ $\sin (x)+\sin (y)=2\sin (\frac{x+y}{2})\cos (\frac{x-y}{2})$ $\sin (x)-\sin (y)=2\cos (\frac{x+y}{2})\sin (\frac{x-y}{2})$ $\cos (x)+\cos (y)=2\cos (\frac{x+y}{2})\cos (\frac{x-y}{2})$ $\cos (x)-\cos (y)=-2\sin (\frac{x+y}{2})\sin (\frac{x-y}{2})$ $2\sin (A)cos(B)=\sin (A+B)+\sin (A-B)$ $2\cos (A)sin(B)=\sin (A+B)-\sin (A-B)$ $2\cos (A)cos(B)=\cos (A+B)+\cos (A-B)$ $2\sin (A)sin(B)=\cos (A-B)-\cos (A+B)$ $\sin (3x)=3\sin (x)-4{\sin }^3(x)$ $\cos (3x)=4{\sin }^3(x)-3\sin (x)$ $\tan (3x)=\frac{3\tan (x)-{\tan }^3(x)}{1-3{\tan }^2(x)}$

Trigonometry General Solutions

Trigonometric Equations	General Solutions
$\sin \theta =0$	$\theta =n\pi$
$\cos \theta =0$	$\theta =(n\pi +\pi /2)$
$\tan \theta =0$	$\theta =n\pi$
$\sin \theta =1$	$\theta =(2n\pi +\pi /2)=(4n+1)\pi /2$
$\cos \theta =1$	$\theta =2n\pi$
$\sin \theta =\sin \alpha$	$\theta =n\pi +(-1{)}^n\alpha \text{ where }\alpha \in [-\pi /2,\pi /2]$
$\cos \theta =\cos \alpha$	$\theta =2n\pi \pm \alpha \text{ where }\alpha \in [0,\pi ]$
$\tan \theta =\tan \alpha$	$\theta =n\pi +\alpha \text{ where }\alpha \in [-\pi /2,\pi /2]$
$\sin 2\theta =\sin 2\alpha$	$\theta =n\pi \pm \alpha$
$\cos 2\theta =\cos 2\alpha$	$\theta =n\pi \pm \alpha$
$\tan 2\theta =\tan 2\alpha$	$\theta =n\pi \pm \alpha$

Perimeters, Areas, Volumes

Shape	TSA Formula	CSA Formula	Volume Formula
Line	N/A	N/A	N/A
Square	$4a$	N/A	N/A
Rectangle	$2(l+b)$	N/A	N/A
Circle	N/A	$2\pi r$	N/A
Triangle	$a+b+c$	N/A	N/A
Cube	$6a^2$	N/A	$a^3$
Cuboid	$2(lb+bh+hl)$	N/A	$l\times b\times h$
Sphere	$4\pi r^2$	$4\pi r^2$	$\frac{4}{3}\pi r^3$
Cylinder	$2\pi r(r+h)$	$2\pi rh$	$\pi r^{2}h$
Cone	$\pi r(r+l)$	$\pi rl$	$\frac{1}{3}\pi r^{2}h$
Hemisphere	$3\pi r^2$	$2\pi r^2$	$\frac{2}{3}\pi r^3$
Rectangular Prism	$2(lw+lh+wh)$	N/A	$l\times w\times h$
Pyramid	$\text{Base Area}+\text{CSA}$	$\frac{1}{2}\times \text{Perimeter}\times l$	$\frac{1}{3}\times \text{Base Area}\times h$
Torus	$4{\pi }^{2}Rr$	N/A	$2{\pi }^{2}R{r}^2$
Ellipse	$2\pi \sqrt{\frac{a^2+b^2}{2}}$	N/A	N/A
Parallelogram	$2(b+h)$	$bh$	N/A
Trapezium	$a+b+c+d$	$\frac{1}{2}(a+b)h$	N/A
Sector	$r^2\theta$	$r\times l$	$\frac{1}{2}{r}^2\theta$
Circular Segment	$r^2\theta -\frac{1}{2}{r}^2\sin (\theta )$	$r\times h$	N/A
Frustum of Cone	$\pi (R+r)(R-r)+\pi R^2+\pi r^2$	$\pi (R+r)l$	$\frac{1}{3}\pi h(R^2+Rr+r^2)$

Integration

General Integrations

$\int a^{x}dx=\frac{a^x}{\ln (a)}+C$ $\int \tan (x)dx=-\ln |\cos (x)|+C$ $\int \cot (x)dx=\ln |\sin (x)|+C$ $\int \sec (x)dx=\ln |\sec (x)+\tan (x)|+C$ $\int \sec (x)dx=-\ln |\sec (x)-\tan (x)|+C$ $\int \csc (x)dx=\ln |\csc (x)-\cot (x)|+C$ $\int \csc (x)dx=-\ln |\csc (x)+\cot (x)|+C$ $\int \frac{f^{\prime }(x)}{f(x)}dx=\ln (f(x))+C$ $\int (f(x){)}^n{f}^{\prime }(x)dx=\frac{(f(x){)}^{n+1}}{n+1}+C$ $\int \frac{f^{\prime }(x)}{\sqrt{f(x)}}dx=2\sqrt{f(x)}+C$

Special Integrations

• general inverse trigonometry substitution results $\int \frac{1}{\sqrt{a^2-x^2}}dx={\sin }^{-1}(\frac{x}{a})+C$ $\int \frac{1}{a^2+x^2}dx=\frac{1}{a}{\tan }^{-1}(\frac{x}{a})+C$ $\int \frac{1}{x\sqrt{x^2-a^2}}dx=\frac{1}{a}{\sec }^{-1}(\frac{x}{a})+C$ $\int \frac{-1}{\sqrt{a^2-x^2}}dx={\cos }^{-1}(\frac{x}{a})+C$ $\int \frac{-1}{a^2+x^2}dx=\frac{1}{a}{\cot }^{-1}(\frac{x}{a})+C$ $\int \frac{-1}{x\sqrt{x^2-a^2}}dx=\frac{1}{a}{\csc }^{-1}(\frac{x}{a})+C$

• general logarithms substitution results $\int \frac{1}{a^2-x^2}dx=\frac{1}{2a}\ln |\frac{x+a}{x-a}|+C$ $\int \frac{1}{x^2-a^2}dx=\frac{1}{2a}\ln |\frac{x-a}{x+a}|+C$ $\int \frac{1}{\sqrt{x^2+a^2}}dx=\ln |x+\sqrt{x^2+a^2}|+C$ $\int \frac{1}{\sqrt{x^2-a^2}}dx=\ln |x+\sqrt{x^2-a^2}|+C$
• inverse and algebra combined general results $\int \sqrt{a^2-x^2}dx=\frac{1}{2}x\sqrt{a^2-x^2}+\frac{1}{2}{a}^2{\sin }^{-1}(\frac{x}{a})+C$ $\int \sqrt{x^2+a^2}dx=\frac{1}{2}x\sqrt{x^2+a^2}+\frac{1}{2}{a}^2\ln |x+\sqrt{x^2+a^2}|+C$ $\int \sqrt{x^2-a^2}dx=\frac{1}{2}x\sqrt{x^2-a^2}-\frac{1}{2}{a}^2\ln |x+\sqrt{x^2-a^2}|+C$

Product Rule Results

$\int e^x(f(x)+f^{\prime }(x))dx=e^{x}f(x)+C$ $\int (x{f}^{\prime }(x)+f(x))dx=xf(x)+C$ $\int e^{ax}cosbxdx=\frac{e^{ax}}{(a^2+b^2)}(acosbx+bsinbx)+C$ $\int e^{ax}sinbxdx=\frac{e^{ax}}{a^2+b^2}(asinbx-bcosbx)+C$ $\int e^{g(x)}(f(x).g^{\prime }(x)+f^{\prime }(x))dx=e^{g(x)}f(x)+C$ $\int \ln (x)dx=x\ln (x)-x+C$ $\int \ln (x)dx=x\ln (x)-x+C$ $\int e^{x^2}dx=e^{x^2}{\sum }_{k=0}^{\infty }(-1{)}^k\frac{d^k(x^2{)}^k}{d{x}^k}=e^{x^2}\left[1-\frac{d(x^2)}{dx}+\frac{d^2(x^2{)}^2}{d{x}^2}-\frac{d^3(x^2{)}^3}{d{x}^3}+...\right]$

Important Substituitions

• Whenever there is a high power inside a power in integrate, take it common, put it equal to t and apply substituion.
• If log is becoming a problem, using substitution and put that logarithm term as t
• Wherever you see $(\sqrt{x^2+1}+x)$ in integration, just put it equal to t and apply substitution.
  • $\sqrt{x^2+1}+x=\frac{1}{\sqrt{x^2+1}-x}$
  • Wherever you see $(\sqrt{x^2+1}-x)$ in integration, just put it equal to 1/t and apply substitution.
• For following integration $\int {\sin }^m(x){\cos }^n(x)dx$
  • If the index of sinx is a positive odd integer, put cosx = t and apply substituition
  • If the index of cosx is a positive odd integer, put sinx = t and apply substituition
• Never fear to put $\frac{linear}{linear}$ = t and apply substitution
• If sin(2x) has cosx - sinx near it , put t = cosx + sinx
• If sin(2x) has cosx + sinx near it , put t = cosx - sinx

$a^2-x^2\text{ or }\sqrt{a^2-x^2}\text{put x}=a\sin \theta \text{ or x =}a\cos \theta$ $a^2+x^2\text{ or }\sqrt{a^2+x^2}\text{put x}=a\tan \theta \text{ or x =}a\cot \theta$ $x^2-a^2\text{ or }\sqrt{x^2-a^2}\text{put x}=a\sec \theta \text{ or x =}a\csc \theta$

• put x = $a\cos (2\theta )$ for following $\sqrt{a+x}\text{ or }\sqrt{a-x}$ $\sqrt{\frac{a+x}{a-x}}\text{ or }\sqrt{\frac{a-x}{a+x}}$
• put x = $a{\cos }^2(\theta )+b{\sin }^2(\theta )$ for following $\sqrt{\frac{x-a}{b-x}}\text{ or }\sqrt{(x-a)(b-x)}$

Integration of Algebric Expressions

$\int \frac{P(x)}{a{x}^2+bx+c}\text{ or }\int \frac{P(x)}{\sqrt{a{x}^2+bx+c}}$

• if P(x) is constant, just complete the square in the denominator
• if P(x) is linear or quadratic then use the algorithm to represent quadratic/linear in form of another
• if degree of P(x) >= 2 then you can also use the devision algorithm to write P(x) in terms of $a{x}^2+bx+c$

  IMPORTANT

  Above point can be used for any algebric expression where degree of numerator is greater than or equal to degree of denominator

$\int \frac{x^2+a^2}{x^4+K{x}^2+a^4}dx\int \frac{x^2-a^2}{x^4+K{x}^2+a^4}dx\int \frac{dx}{x^4+K{x}^2+a^4}dx\int \frac{x^{2}dx}{x^4+K{x}^2+a^4}dx$ For above integrals, devide both numerator and denominator by $x^2$

$1-\frac{1}{x^2}=\frac{d}{dx}(x+\frac{1}{x})$ $1+\frac{1}{x^2}=\frac{d}{dx}(x-\frac{1}{x})$ $x^2+\frac{1}{x^2}=(x+\frac{1}{x}{)}^2-2$ $x^2+\frac{1}{x^2}=(x-\frac{1}{x}{)}^2+2$

• If you want to represent a linear equation in form of derivative of another quadratic, you can use following formula $ax+b=\alpha (\frac{d(p{x}^2+qx+r)}{dx})+\beta$

After simplifying above equations, just compare the coefficients of various terms for find out value of the contstants alpha, beta and lambda

Integration by Partial Fractions

$\frac{P(x)}{(x-a_1)(x-a_2)}=\frac{A}{x-a_1}+\frac{B}{x-a_2}$ $\frac{P(x)}{(x-a_1{)}^2(x-a_2)}=\frac{A}{x-a_1}+\frac{B}{(x-a_1{)}^2}+\frac{C}{x-a_2}$ $\frac{P(x)}{(a{x}^2+bx+c)(x-a_1)}=\frac{Ax+B}{a{x}^2+bx+c}+\frac{C}{x-a_1}$ $\frac{P(x)}{(a{x}^2+bx+c{)}^2(x-a_1{)}^2}=\frac{Ax+B}{a{x}^2+bx+c}+\frac{Cx+D}{(a{x}^2+bx+c{)}^2}+\frac{E}{x-a_1}+\frac{F}{(x-a_1{)}^2}$

Integration By Parts

$\int f(x)g(x)dx=f(x)\int g(x)dx-\int [\frac{df(x)}{dx}\times \int g(x)dx]dx$

• Function whose integration is easy, is taken as the second function
• If integrad contains only 1 function which cannot be integrated directory, then we take second function as 1 and try By Parts
• Usually, first function is taken in order of ILATE i.e inverse function, log function, algebric function, trigonometric function and then exponential function

Inverse Trigonometry

Domain-Range Table

Here, family 1,2 are only used in order to group functions with similar domain and range together for better understanding.

Family 1			Family 2
Function	Domain	Range	Function	Domain	Range
${\sin }^{-1}(x)$	$[-1,1]$	$\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$	${\cos }^{-1}(x)$	$[-1,1]$	$[0,\pi ]$
${\csc }^{-1}(x)$	$\mathbb{R}-(-1,1)$	$\left[-\frac{\pi }{2},\frac{\pi }{2}\right]-\{0\}$	${\sec }^{-1}(x)$	$\mathbb{R}-(-1,1)$	$[0,\pi ]-\{\frac{\pi }{2}\}$
${\tan }^{-1}(x)$	$\mathbb{R}$	$\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$	${\cot }^{-1}(x)$	$\mathbb{R}$	$(0,\pi )$

Inverse Trigonometric function of -x

Family 1	Family 2
${\sin }^{-1}(-x)=-{\sin }^{-1}(x)$	${\cos }^{-1}(-x)=\pi -{\cos }^{-1}(x)$
${\csc }^{-1}(-x)=-{\csc }^{-1}(x)$	${\sec }^{-1}(-x)=\pi -{\sec }^{-1}(x)$
${\tan }^{-1}(-x)=-{\tan }^{-1}(x)$	${\cot }^{-1}(-x)=\pi -{\cot }^{-1}(x)$

ITF Conversion Formulas

${\sin }^{-1}(\frac{1}{x})={\csc }^{-1}(x)x\ge 1\text{OR}x\le -1$ ${\cos }^{-1}(\frac{1}{x})={\sec }^{-1}(x)x\ge 1\text{OR}x\le -1$ ${\sec }^{-1}(\frac{1}{x})={\cos }^{-1}(x)x\ge 1\text{OR}x\le -1$ ${\cot }^{-1}(\frac{1}{x})={\tan }^{-1}(x)x>0$ ${\tan }^{-1}(\frac{1}{x})={\cot }^{-1}(x)x>0$ ${\tan }^{-1}(\frac{1}{x})=-\pi +{\cot }^{-1}(x)x<0$

Single Angle, Multiple ITC Formulas

${\sin }^{-1}(x)+{\cos }^{-1}(x)=\frac{\pi }{2}x\in [-1,1]$ ${\tan }^{-1}(x)+{\cot }^{-1}(x)=\frac{\pi }{2}x\in \mathbb{R}$ ${\csc }^{-1}(x)+{\sec }^{-1}(x)=\frac{\pi }{2}|x|\ge 1$

Multiple Angle Formulas

${\tan }^{-1}(x)+{\tan }^{-1}(y)={\tan }^{-1}(\frac{x+y}{1-xy})xy<1$ ${\tan }^{-1}(x)+{\tan }^{-1}(y)=\frac{\pi }{2}xy=1$ ${\tan }^{-1}(x)+{\tan }^{-1}(y)=\pi +{\tan }^{-1}(\frac{x+y}{1-xy})xy>1$ ${\sin }^{-1}(x)+{\sin }^{-1}(y)={\sin }^{-1}(x\sqrt{1-y^2}+y\sqrt{1-x^2})x^2+y^2\le 1$ ${\sin }^{-1}(x)+{\sin }^{-1}(y)=\pi -{\sin }^{-1}(x\sqrt{1-y^2}+y\sqrt{1-x^2})x^2+y^2\ge 1$ ${\sin }^{-1}(x)-{\sin }^{-1}(y)={\sin }^{-1}(x\sqrt{1-y^2}-y\sqrt{1-x^2})x\ge 0,y\ge 0$ ${\cos }^{-1}(x)-{\cos }^{-1}(y)={\cos }^{-1}(xy+\sqrt{1-x^2}\cdot \sqrt{1-y^2})x\le y$ ${\cos }^{-1}(x)-{\cos }^{-1}(y)=-{\cos }^{-1}(xy+\sqrt{1-x^2}\cdot \sqrt{1-y^2})x\ge y$

ITF Multiple Formulas

$2{\tan }^{-1}(x)={\tan }^{-1}\left(\frac{2x}{1-x^2}\right)\text{for}|x|<1$ $2{\tan }^{-1}(x)=\pi +{\tan }^{-1}\left(\frac{2x}{1-x^2}\right)\text{for}x>1$ $2{\tan }^{-1}(x)=-\pi +{\tan }^{-1}\left(\frac{2x}{1-x^2}\right)\text{for}x<-1$ $2{\tan }^{-1}(x)={\sin }^{-1}\left(\frac{2x}{1+x^2}\right)\text{for}|x|\le 1$ $2{\tan }^{-1}(x)=\pi +{\sin }^{-1}\left(\frac{2x}{1+x^2}\right)\text{for}x>1$ $2{\tan }^{-1}(x)=-\pi +{\sin }^{-1}\left(\frac{2x}{1+x^2}\right)\text{for}x<-1$ $2{\tan }^{-1}(x)={\cos }^{-1}\left(\frac{1-x^2}{1+x^2}\right)\text{for}x\ge 0$ $2{\tan }^{-1}(x)=-{\cos }^{-1}\left(\frac{1-x^2}{1+x^2}\right)\text{for}x\le 0$ $3{\tan }^{-1}(x)={\tan }^{-1}\left(\frac{3x-x^3}{1-3x^2}\right)\text{for}\frac{-1}{\sqrt{3}}<x<\frac{1}{\sqrt{3}}$ $3{\tan }^{-1}(x)=\pi +{\tan }^{-1}\left(\frac{3x-x^3}{1-3x^2}\right)\text{for}x>\frac{1}{\sqrt{3}}$ $3{\tan }^{-1}(x)=-\pi +{\tan }^{-1}\left(\frac{3x-x^3}{1-3x^2}\right)\text{for}x<\frac{-1}{\sqrt{3}}$

Limits and Derivatives

Limits

${\lim }_{x\to 0}\frac{\sin (x)}{x}={\lim }_{x\to 0}\frac{1-\cos (x)}{x}={\lim }_{x\to 0}\frac{\tan (x)}{x}=1$ ${\lim }_{x\to 0}\frac{{\sin }^{-1}(x)}{x}={\lim }_{x\to 0}\frac{{\tan }^{-1}(x)}{x}=1$ ${\lim }_{x\to 0}\frac{e^x-1}{x}={\lim }_{x\to \infty }\frac{\ln (1+x)}{x}=1$ ${\lim }_{x\to 0}(1+x{)}^{1/x}={\lim }_{x\to \infty }(1+\frac{1}{x}{)}^x=e$ ${\lim }_{x\to 0}\frac{a^x-1}{x}=\ln (a)$ ${\lim }_{x\to \infty }(1+\frac{a}{x}{)}^x=e^a$ ${\lim }_{x\to a}\frac{x^n-a^n}{x-a}=n{a}^{n-1}$ ${\lim }_{n\to \infty }\frac{t^n-1}{t^n+1}=1$ ${\lim }_{x\to a}(1+f(x){)}^{g(x)}=e^{{\lim }_{x\to a}(f(x)\times g(x))}$

Derivatives

$\frac{d}{dx}[{\sin }^{-1}x]=\frac{1}{\sqrt{1-x^2}}$ $\frac{d}{dx}[{\cos }^{-1}x]=\frac{-1}{\sqrt{1-x^2}}$ $\frac{d}{dx}[{\tan }^{-1}x]=\frac{1}{x^2+1}$ $\frac{d}{dx}[{\cot }^{-1}x]=\frac{-1}{x^2+1}$ $\frac{d}{dx}[{\sec }^{-1}x]=\frac{1}{|x|\sqrt{x^2-1}}$ $\frac{d}{dx}[{\csc }^{-1}x]=\frac{-1}{|x|\sqrt{x^2-1}}$

Matrices and Determinants

Types of Matrices

1. Row Matrix: A matrix with only 1 row.
2. Column Matrix: A matrix with only 1 column.
3. Square Matrix: A matrix with an equal number of rows and columns.
4. Rectangular Matrix: A matrix with an unequal number of rows and columns.
5. Zero Matrix: A matrix where all elements are zero, regardless of size.
6. Diagonal Matrix: A square matrix where all elements except for the main diagonal are zero.
7. Scalar Matrix: A square matrix with a constant value on the diagonal and zeros elsewhere.
8. Unity Matrix: A matrix where all elements are 1, regardless of size.
9. Identity Matrix: A square matrix with 1s along the main diagonal and 0s elsewhere.
10. Upper Triangular Matrix: A square matrix where all elements below the diagonal are zero.
11. Lower Triangular Matrix: A square matrix where all elements above the diagonal are zero.
12. Symmetric Matrix: A square matrix that is equal to its transpose.
13. Skew-Symmetric Matrix: A square matrix that is equal to the negative of its transpose.
14. Non-Singular (Invertible) Matrix: A square matrix with a non-zero determinant.
15. Singular (Non-Invertible) Matrix: A square matrix with a determinant of zero.
16. Involutory Matrix: A square matrix that is its own inverse, meaning $A^2=I$, where $I$ is the identity matrix.
17. Orthogonal Matrix: A square matrix such that $A{A}^T=I$.
    • Important: Its determinant is $\pm 1$.
    • Conditions:
      • Magnitude of every column must be 1.
      • Dot product of columns must be zero.
18. Idempotent Matrix: A square matrix such that $A^n=A$ for every $n\ge 2$.
19. Matrix with $A^2=I$: Its determinant is either 1 or -1.
20. Inverse Matrix: A square matrix B is called the inverse of A if $AB=BA=I$

Key Points about Matrices

1. Left Distribution Law: $A(B+C)=AB+AC$.
2. Right Distribution Law: $(B+C)A=BA+CA$.
3. If $A+A^T$ is symmetric, then $A-A^T$ is skew-symmetric.
4. Any square matrix can be expressed as a sum of a symmetric and skew-symmetric matrix: $A=\frac{1}{2}(A+A^T)+\frac{1}{2}(A-A^T)$
   • Here, $P=\frac{1}{2}(A+A^T)$ (symmetric) and $Q=\frac{1}{2}(A-A^T)$ (skew-symmetric).
5. The product $AB$ is symmetric if and only if $AB=BA$.
6. The sum $AB+BA$ is symmetric, while $AB-BA$ is skew-symmetric.
7. $(A^T{)}^n=(A^n{)}^T$.

Properties of Determinants

1. The value of a determinant remains unchanged if its rows and columns are interchanged.
2. If any two rows (or columns) of a determinant are interchanged, the sign of the determinant changes.
3. If any two rows (or columns) of a determinant are identical, the value of the determinant is zero.
4. If all elements of a row (or column) are multiplied by a scalar $K$, the value of the determinant is also multiplied by $K$: $K\times \left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|=\left|\begin{array}{ccc}K{a}_1& K{b}_1& K{c}_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|$
5. If some or all elements of a row (or column) are expressed as the sum of two (or more) terms, the determinant can be expressed as the sum of two (or more) determinants of the same order: $\left|\begin{array}{ccc}{a}_1+x& b_1+y& c_1+z\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|=\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|+\left|\begin{array}{ccc}x& y& z\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|$
6. The value of the determinant is not altered by adding or subtracting a multiple of any row (or column) to another row (or column): $\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|=\left|\begin{array}{ccc}{a}_1+K{a}_3& b_1+K{b}_3& c_1+K{c}_3\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|$
7. Factor Theorem: If substituting $x=a$ makes the determinant vanish, then $(x-a)$ is a factor of the determinant.

Cofactor Determinant

The determinant formed by replacing all elements of a determinant with their respective cofactors is called the cofactor determinant. If $|A{|}_C$ is the cofactor determinant of $A$, then: $|A|\_C=|A{|}^{n-1}$

Determinants Key Points

1. Determinant operations always yield scalar values.
2. If $k$ is any scalar and $A$ is any matrix, then $|kA|=k^n|A|$, where $n$ is the order of matrix $A$.
3. The area of a triangle with vertices $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$ can be found using the determinant: $\text{Area}=\frac{1}{2}\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|$
4. The inverse of a square matrix $A$ (if it exists) is given by: $A^{-1}=\frac{\text{adj}(A)}{|A|}\text{if }|A|\ne 0$
5. The product of a matrix and its adjugate is: $A\cdot \text{adj}(A)=\text{adj}(A)\cdot A=|A|I$
6. The product of a matrix and its inverse is: $A{A}^{-1}=I=A^{-1}A$
7. The inverse $A^{-1}$ exists only when $A$ is a non-singular matrix.
8. The inverse of a square matrix, if it exists, is unique.
9. If $A$ is a non-singular matrix, then both $A^{-1}$ and $A^T$ are also non-singular.
10. The determinant of a skew-symmetric matrix of odd order is zero.
11. The determinant of a skew-symmetric matrix of even order is a non-zero perfect square.
12. For a square matrix A, $|A^T|=|A|$.

Determinant Results

$|AB|=|A|\times |B|$ $(A^{-1}{)}^{-1}=A$ $(AB{)}^{-1}=B^{-1}{A}^{-1}$ $(A^T{)}^{-1}=(A^{-1}{)}^T$ $|\text{adj}(A)|=|A{|}^{n-1}$ $|A^{-1}|=|A{|}^{-1}$ $\text{adj}(AB)=\text{adj}(B)\cdot \text{adj}(A)$ $\text{adj}(A^m)=\text{adj}(A{)}^m$ $\text{adj}(kA)=k^{n-1}\text{adj}(A)$ $\text{adj}(A^T)=(\text{adj}(A){)}^T$ $\text{adj}(\text{adj}(A))=|A{|}^{n-2}A$ $\text{adj}(\text{adj}(A))|=|A{|}^{(n-1{)}^2}$