First Year (FE)
- What is a Jacobian of two or three variables and how do I calculate it? In one line The Jacobian J=∂(u,v)/∂(x,y) is the determinant of all first partial derivatives of u,v with respect to x,y ; it measures how a... 1
- What is Lagrange's method of undetermined multipliers for finding maxima and minima with a constraint? In one line To optimise f(x,y,z) subject to φ(x,y,z)=0 , form F=f+λφ and solve F_x=F_y=F_z=0 together with φ=0 . Explained simply Suppose yo... 1
- What are symmetric and skew-symmetric matrices, and how is any square matrix written as their sum? In one line A square matrix is symmetric if A^T=A and skew-symmetric if A^T=-A ; every square matrix equals 1/2(A+A^T)+1/2(A-A^T) , a symmet... 1
- How do I find the inverse of a 3×3 matrix using the adjoint method? In one line If |A|≠0 , then A⁻¹=1/|A| adj A , where adj A is the transpose of the matrix of cofactors. Explained simply The inverse of a num... 1
- What are Hermitian and skew-Hermitian matrices and what properties do they have? In one line A complex square matrix is Hermitian if it equals its conjugate transpose ( A^θ=A ) and skew-Hermitian if A^θ=-A ; they are the... 1
- What are orthogonal and unitary matrices and how do I check whether a matrix is one? In one line A real matrix is orthogonal if AA^T=I (so A⁻¹=A^T ); a complex matrix is unitary if AA^θ=I (so A⁻¹=A^θ ). Explained simply Rotat... 1
- What is the rank of a matrix and how do elementary transformations help find it? In one line The rank of a matrix is the order of its largest non-zero minor, which equals the number of linearly independent rows; elementar... 1
- How do I find the rank of a matrix by reducing it to echelon form? In one line Use row operations to make the matrix upper-triangular in a staircase pattern (echelon form); the rank is the number of non-zero... 1
- What is the normal form of a matrix and how do I reduce a matrix to it to find rank? In one line The normal form of a matrix is beginbmatrixI_r&0\\0&0endbmatrix , obtained with both row and column operations; the size r of th... 1
- How do I find non-singular matrices P and Q such that PAQ is in normal form? In one line Write A=I A I ; every row operation on A is also done on the left identity (building P ) and every column operation on the right... 1
- When does a homogeneous system AX = 0 have non-trivial solutions, and how do I find them? In one line AX=0 always has the trivial solution X=0 ; it has non-trivial (infinitely many) solutions exactly when ρ(A)<n , the number of un... 1
- How do I test a non-homogeneous system AX = B for consistency and find its solutions? In one line Compare the rank of A with the rank of the augmented matrix [A | B] : unequal ranks mean no solution; equal ranks =n mean a uniq... 1
- How is the inverse of a matrix used to encode and decode messages in coding theory? In one line A message is turned into number columns, scrambled by multiplying with an invertible matrix A , and unscrambled by multiplying w... 1
- What is a transcendental equation and how do I locate an interval containing its root? In one line A transcendental equation contains non-polynomial functions such as e^x , log x , sin x (e.g. x=cos x ); we locate a root by fin... 1
- How does the Newton–Raphson method find a root of f(x) = 0 and why does it work? In one line Newton–Raphson improves a guess x_n by sliding down the tangent line to the x -axis: x_(n+1)=x_n-f(x_n)/f'(x_n) . Explained simp... 1
- How do I derive Newton–Raphson iterative formulas for square roots, cube roots and reciprocals? In one line Write the required number as the root of a simple equation, such as x²-N=0 for sqrt N , and apply x_(n+1)=x_n-f(x_n)/f'(x_n) to... 1
- How does the Regula Falsi (false position) method find a root, and how is it different from Newton–Raphson? In one line Regula Falsi joins the points (a,f(a)) and (b,f(b)) on opposite sides of the axis by a straight chord and takes where the chord... 1
- How does the Gauss–Jacobi iteration method solve a system of linear equations? In one line Gauss–Jacobi rewrites each equation to make one unknown the subject, then repeatedly substitutes the previous round's values int... 1
- How is the Gauss–Seidel method different from Gauss–Jacobi, and why does it converge faster? In one line Gauss–Seidel is Jacobi with one change: each newly computed value is used immediately in the following equations of the same ite... 1
- What is diagonal dominance and why must equations be rearranged before Jacobi or Seidel iteration? In one line A system is (strictly) diagonally dominant when, in every row, the size of the diagonal coefficient is greater than the sum of t... 1
- What are indeterminate forms 0/0 and ∞/∞, and how does L'Hospital's rule evaluate them? In one line When a limit takes the form 0/0 or frac∞∞ , L'Hospital's rule says limf(x)/g(x)=limf'(x)/g'(x) , provided the new limit exists.... 1
- How do I evaluate other indeterminate forms like 0·∞, ∞−∞, 1^∞, 0⁰ and ∞⁰? In one line Convert them to 0/0 or frac∞∞ : rewrite products and differences as a single fraction, and for powers take logarithms first; the... 1
- How does Gauss elimination solve a system of linear equations step by step? In one line Gauss elimination uses row operations to turn the augmented matrix into upper-triangular form, then finds the unknowns by back-s... 1
- How does the Gauss–Jordan method solve linear equations and how is it different from Gauss elimination? In one line Gauss–Jordan continues the elimination above the pivots as well, reducing the coefficient matrix to the identity so that the sol... 1