The transpose of a matrix A, denoted A' (or A^T), is the matrix obtained by interchanging its rows and columns. Two properties are: (1) (A')' = A, that is, the transpose of the transpose is the original matrix; and (2) (AB)' = B' A', the transpose of a product equals the product of the transposes in reverse order. Also (A + B)' = A' + B' and (kA)' = k A'.

(AB)' = B' A'

A square matrix A is symmetric if A' = A, that is, aij = aji for all i and j (it is unchanged on taking its transpose). A square matrix A is skew-symmetric if A' = -A, that is, aij = -aji for all i and j; this forces all its diagonal elements to be zero. Every square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix.

symmetric: A' = A; skew-symmetric: A' = -A

If a matrix has 12 elements, then the order (number of rows x number of columns) must be a pair of factors whose product is 12. The possible orders are: 1 x 12, 12 x 1, 2 x 6, 6 x 2, 3 x 4 and 4 x 3. So there are 6 possible orders. (In general, the number of possible orders equals the number of ways of writing the total number of elements as a product of two natural numbers.)

First find 2A = [[2, 4],[6, 8]]. Then subtract B element by element: 2A - B = [[2 - 2, 4 - 1],[6 - 0, 8 - 5]] = [[0, 3],[6, 3]].

Expand along the first row: |A| = 1 x (1 x 0 - 4 x 6) - 2 x (0 x 0 - 4 x 5) + 3 x (0 x 6 - 1 x 5) = 1 x (0 - 24) - 2 x (0 - 20) + 3 x (0 - 5) = -24 + 40 - 15 = 1. So the value of the determinant is 1.

(1) The value of a determinant remains unchanged if its rows and columns are interchanged (that is, |A'| = |A|). (2) If any two rows (or two columns) of a determinant are interchanged, the sign of the determinant changes. (3) If any two rows (or columns) of a determinant are identical (or proportional), the value of the determinant is zero. Also, if each element of a row (or column) is multiplied by a constant k, the determinant is multiplied by k.

The minor of an element aij of a determinant is the value of the determinant obtained by deleting the ith row and the jth column in which that element lies; it is denoted by Mij. The cofactor of the element aij is Aij = (-1)^(i + j) x Mij, that is, the minor with a sign attached according to the position of the element. Cofactors are used to expand a determinant and to find the adjoint of a matrix.

Aij = (-1)^(i+j) Mij

A square matrix A is called singular if its determinant is zero (|A| = 0), and non-singular if its determinant is non-zero (|A| is not equal to 0). The condition for a square matrix to be invertible (to have an inverse) is that it must be non-singular, that is, |A| must not be zero, because the inverse A inverse = (1/|A|) adj A is defined only when |A| is not zero.

A inverse exists if and only if |A| is not 0

For an n x n matrix, |adj A| = |A|^(n - 1). Here n = 3, so |adj A| = |A|^2 = 5^2 = 25. Also, for a scalar k, |kA| = k^n |A|. Here |2A| = 2^3 x |A| = 8 x 5 = 40. So |adj A| = 25 and |2A| = 40.

|adj A| = |A|^(n-1); |kA| = k^n |A|

A function f(x) is continuous at a point x = a if the following three conditions are all satisfied: (1) f(a) is defined (the function has a value at a); (2) the limit of f(x) as x approaches a exists (that is, the left-hand limit equals the right-hand limit); and (3) this limit is equal to the value of the function, that is, limit of f(x) as x approaches a = f(a).

limit(x->a) f(x) = f(a)

If a function is differentiable at a point, then it is necessarily continuous at that point. However, the converse is not always true: a function that is continuous at a point need not be differentiable there. For example, f(x) = |x| is continuous at x = 0 but not differentiable at x = 0 (it has a sharp corner). So differentiability implies continuity, but continuity does not imply differentiability.

Let y = sin(x^2). Using the chain rule, dy/dx = cos(x^2) x d/dx(x^2) = cos(x^2) x 2x = 2x cos(x^2). The chain rule states that the derivative of a composite function f(g(x)) is f'(g(x)) x g'(x).

d/dx f(g(x)) = f'(g(x)) g'(x)

The derivative of sin inverse (x) with respect to x is 1/sqrt(1 - x^2), valid for -1 < x < 1. The derivative of tan inverse (x) with respect to x is 1/(1 + x^2), valid for all real x. These standard results are obtained by implicit differentiation of y = sin inverse (x) and y = tan inverse (x).

d/dx(sin inverse x) = 1/sqrt(1 - x^2)

Differentiate each with respect to the parameter t: dx/dt = -a sin t and dy/dt = a cos t. Then dy/dx = (dy/dt)/(dx/dt) = (a cos t)/(-a sin t) = -cos t/sin t = -cot t. So dy/dx = -cot t.

dy/dx = (dy/dt)/(dx/dt)

Rolle's theorem: if a function f is continuous on the closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one point c in (a, b) such that f'(c) = 0. Lagrange's mean value theorem: if a function f is continuous on [a, b] and differentiable on (a, b), then there exists at least one point c in (a, b) such that f'(c) = (f(b) - f(a))/(b - a).

f'(c) = (f(b) - f(a))/(b - a)

The derivative dy/dx represents the instantaneous rate of change of y with respect to x. When a quantity changes with time, its derivative with respect to time gives the rate of change. The area of a circle is A = pi r^2, so the rate of change of area with respect to the radius is dA/dr = 2 pi r. If the radius changes with time, dA/dt = 2 pi r (dr/dt).

dA/dr = 2 pi r

A differentiable function f is increasing on an interval if its derivative f'(x) is greater than or equal to 0 (strictly increasing if f'(x) > 0) for all x in that interval. It is decreasing if its derivative f'(x) is less than or equal to 0 (strictly decreasing if f'(x) < 0) for all x in that interval. So the sign of the first derivative tells us whether the function rises or falls.

f'(x) > 0 increasing; f'(x) < 0 decreasing

The slope of the tangent to a curve y = f(x) at a point is the value of the derivative dy/dx at that point. The normal is the line perpendicular to the tangent at that point, so its slope is the negative reciprocal of the tangent's slope, that is, slope of normal = -1/(dy/dx). Using the point-slope form with these slopes gives the equations of the tangent and the normal.

slope of normal = -1/(slope of tangent)

Let f be a function and c be a point where f'(c) = 0 (a critical point). By the second derivative test: if f''(c) < 0, then f has a local maximum at c; if f''(c) > 0, then f has a local minimum at c; and if f''(c) = 0, the test fails and we must use the first derivative test (or higher derivatives) to decide.

Integration is the reverse (inverse) process of differentiation; it is used to find a function whose derivative is given. The indefinite integral of a function includes an arbitrary constant of integration C. The power rule for integration is: the integral of x^n dx = x^(n + 1)/(n + 1) + C, valid for n not equal to -1. For n = -1, the integral of (1/x) dx = ln|x| + C.

integral of x^n dx = x^(n+1)/(n+1) + C