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

Integrate term by term using the power rule. The integral of x^3 dx = x^4/4, the integral of 2x dx = x^2, and the integral of 1 dx = x. Adding a single constant of integration, the integral of (x^3 + 2x + 1) dx = x^4/4 + x^2 + x + C.

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

The standard integrals are: the integral of sin x dx = -cos x + C; the integral of cos x dx = sin x + C; and the integral of e^x dx = e^x + C. Also, the integral of sec^2 x dx = tan x + C. These are obtained by reversing the corresponding derivative formulas.

integral of sin x dx = -cos x + C

(1) The integral from a to b of f(x) dx = - the integral from b to a of f(x) dx (interchanging the limits changes the sign). (2) The integral from a to b of f(x) dx = the integral from a to c of f(x) dx + the integral from c to b of f(x) dx (splitting the interval at a point c). (3) The integral from 0 to a of f(x) dx = the integral from 0 to a of f(a - x) dx. Also, for an odd function the integral from -a to a is 0, and for an even function it is twice the integral from 0 to a.

integral(a to b) f(x) dx = -integral(b to a) f(x) dx

Integrate term by term. The integral of (1/x) dx = ln|x| and the integral of e^x dx = e^x. Adding the constant of integration, the integral of (1/x + e^x) dx = ln|x| + e^x + C.

integral of (1/x) dx = ln|x| + C

The area bounded by the curve y = f(x), the x-axis, and the ordinates x = a and x = b (with f(x) >= 0) is given by the definite integral: Area = the integral from a to b of y dx = the integral from a to b of f(x) dx. The area between two curves y = f(x) (upper) and y = g(x) (lower) from x = a to x = b is Area = the integral from a to b of [f(x) - g(x)] dx.

Area = integral(a to b) f(x) dx

If a region is symmetric about the x-axis, the y-axis or the origin, we can compute the area of one symmetric part and then multiply by the appropriate factor to get the total area. For example, a circle x^2 + y^2 = a^2 is symmetric about both axes, so its total area equals 4 times the area of the part lying in the first quadrant. Using symmetry reduces the amount of integration needed and simplifies the calculation.

A differential equation is an equation that involves an independent variable, a dependent variable and the derivatives of the dependent variable with respect to the independent variable. The order of a differential equation is the order of the highest derivative appearing in it. The degree of a differential equation is the power (exponent) of the highest order derivative, provided the equation is a polynomial in the derivatives (free from radicals and fractions in the derivatives).