GATE Engineering Mathematics formula sheet

The order reverses for any number of factors, e.g. $(ABC{)}^{-1}=C^{-1}{B}^{-1}{A}^{-1}$. The transpose and conjugate-transpose laws hold whenever $AB$ is defined, square or not; the inverse laws need every factor square and non-singular.

$A$ non-singular, $k\ne 0$ a scalar and $m$ a positive integer. A diagonal matrix with non-zero diagonal entries $d_1,\dots ,d_n$ has inverse $\mathrm{diag}(1/d_1,\dots ,1/d_n)$, and the inverse of a non-singular upper (lower) triangular matrix is again upper (lower) triangular.

$x$ in radians, $a>0$. Each follows from the Maclaurin series of the function.

The last holds for any real $n$ when $a>0$.

A scalar; $\theta \in [0,\pi ]$ is the angle between the vectors. Commutative. For non-zero vectors, $\mathbf{A}\cdot \mathbf{B}=0$ means perpendicular. $\hat{i}\cdot \hat{i}=1$, $\hat{i}\cdot \hat{j}=0$.

The scalar component of $\mathbf{A}$ along $\mathbf{B}$ is signed (negative when $\theta >90^{\circ }$); the vector projection lies along $\pm \mathbf{B}$. The direction cosines of a non-zero $\mathbf{A}$ are $A_x/|\mathbf{A}|$, $A_y/|\mathbf{A}|$, $A_z/|\mathbf{A}|$, and their squares add to 1.

Linear means $y$ and every derivative appear only to the first power, are never multiplied together, and never sit inside another function. The coefficients may depend on $x$ only, and the equation has order $n$ wherever $a_0(x)\ne 0$. The equation is homogeneous when $r(x)=0$ and non-homogeneous otherwise.

Valid where $g(y)\ne 0$. Each root $y=y_0$ of $g(y)=0$ is a separate constant solution, and dividing by $g(y)$ can lose it (sometimes it reappears for a particular value of the constant). For $\frac{dy}{dx}=f(ax+by+c)$, substitute $z=ax+by+c$ to get the separable equation $\frac{dz}{dx}=a+bf(z)$.

Conjugate $\bar{z}=x-jy=r{e}^{-j\theta }$, with $z\bar{z}=|z{|}^2$, $\mathrm{Re}z=\frac{z+\bar{z}}{2}$ and $\mathrm{Im}z=\frac{z-\bar{z}}{2j}$. The argument is fixed only up to adding $2k\pi$.

Here ${\tan }^{-1}$ returns a value in $(-\pi /2,\pi /2)$, and $\mathrm{Arg}0$ is undefined. Some texts take the principal range as $[0,2\pi )$; these notes use $(-\pi ,\pi ]$ throughout, so $\mathrm{Arg}(-1)=\pi$.

Holds for any two events. Use the complement for every 'at least one' question: $P(\text{at least one})=1-P(\text{none})$.

Add singles, subtract pairs, add the triple. The same pattern counts elements of a union: replace $P$ by $n$.

Doolittle form: $L$ is unit lower triangular (1s on its diagonal) and holds the multipliers $m_{ik}$ below the diagonal; $U$ is the upper-triangular matrix that Gaussian elimination without row swaps produces. Crout form puts the 1s on the diagonal of $U$ instead. For a non-singular $A$, the factorisation exists without row swaps exactly when every leading principal minor of $A$ is non-zero. One factorisation serves any number of right-hand sides $b$.

Every new component uses only the previous sweep's values. Needs $a_{ii}\ne 0$ (reorder the equations if necessary).