Catalonia PAU Applied Mathematics for Social Sciences Exam key terms

Matrix obtained by turning rows into columns and columns into rows; entry (i,j) of the transpose equals entry (j,i) of the original matrix.

Square matrix that equals its own transpose, so entries mirrored across the main diagonal are identical; A = Aᵗ.

Matrix B such that A·B = B·A = I, the identity. It exists only for square matrices with non-zero determinant, and then B = adj(A)/det(A), with adj(A) defined as the already-transposed cofactor matrix.

Scalar assigned to a square matrix that is zero exactly when the matrix is singular; computed by cofactor expansion or by row operations that track how the value changes.

Determinant of the submatrix left after deleting one row and one column of a square matrix; the associated cofactor multiplies it by (−1)^(i+j).

Transpose of the cofactor matrix of A; dividing it by the determinant of A produces the inverse matrix when that determinant is non-zero.

Maximum number of linearly independent rows, equal to the maximum number of independent columns; found by reducing the matrix to echelon form and counting non-zero rows.

Rule classifying a linear system by comparing the rank of the coefficient matrix with the rank of the augmented matrix and with the number of unknowns.

Linear system with at least one solution and exactly one; occurs when the ranks of coefficient and augmented matrices both equal the number of unknowns.

Linear system with infinitely many solutions; the common rank of coefficient and augmented matrices is smaller than the number of unknowns.

Linear system with no solution at all, detected when the rank of the augmented matrix exceeds the rank of the coefficient matrix.

Letter appearing in the coefficients or constants of a system; the discussion studies for which values the system is compatible and how many solutions it has.

Value L that f(x) approaches as x approaches a point; it may exist even when the function is undefined or takes another value at that point.

Property of a function at a point where the limit exists, the function is defined there, and both values coincide; differentiability implies it but not conversely.

Expression such as 0/0, ∞/∞, ∞−∞ or 0·∞ whose limit cannot be read off directly and requires algebraic manipulation, L'Hôpital's rule or equivalent techniques.

Method for indeterminate quotients: replace the limit of f/g by the limit of the quotient of derivatives f'/g', provided the original quotient has form 0/0 or ∞/∞ and the theorem’s differentiability, nonzero denominator-derivative and derivative-limit hypotheses hold.

Limit of the difference quotient, giving the slope of the tangent line and the instantaneous rate of change of a function at a point.

Existence of the derivative at a point; it requires continuity there, and at a breakpoint the two one-sided derivatives must agree.

Line touching a curve at a point with slope equal to the derivative there; its equation is y − f(a) = f'(a)(x − a).

Search for maxima or minima of a function, usually by solving f'(x) = 0 and classifying the critical points with the sign of the second derivative or a sign table.

If a function is continuous on a closed interval and takes values of opposite sign at its ends, then it vanishes at some interior point; used to locate roots.

For a function continuous on [a,b] and differentiable on (a,b) with equal endpoint values, there is an interior point where the derivative is zero.

Selections of k elements from n without regard to order, counted by C(n,k) = n!/(k!(n−k)!); used when only the group matters.

Ordered arrangements of all or part of a set of elements; the number of orderings of n distinct items is n!, and repeated items require division by their factorials.

Probability of A given that B has occurred, P(A|B) = P(A∩B)/P(B), defined when B has non-zero probability.

Two events are independent when P(A∩B) = P(A)·P(B), so knowing one occurred does not change the probability of the other.

Formula reversing a conditional probability, P(A|B) = P(B|A)P(A)/P(B), with the denominator expanded over a partition of the sample space.

Graph whose branches represent successive stages of an experiment with probabilities on the edges; path products give intersections and sums give totals.

Model for the number of successes in n independent trials with constant success probability p; probabilities come from C(n,k)pᵏ(1−p)ⁿ⁻ᵏ, with mean np.

Continuous symmetric bell-shaped distribution fixed by mean μ and standard deviation σ; probabilities are areas under the curve found by standardising to N(0,1).