The Inverse of A Matrix

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Let A  be an n*n  matrix. An n*n  matrix A-1  such that                                                                                      AA-1=A-1A=In is the inverse of A. A matrix A  is nonsingular if A-1exists (i.e., if A has an inverse). If a matrix does not have an inverse, then it is singular. Examples Explanation                 Calculating The Inverse Given a square matrix M, we know the size of its inverse (the same size as M) and the product of M  and its inverse. Using this information, the inverse of M  can be calculated by assigning variables to the elements of M -1 and representing the product MM -1 as a system of n  equations in n  unknowns, where M  has dimension n*n. Examples Explanation     This system can be expressed as an augmented matrix. Recall that an augmented matrix includes the constant coefficients of a system of equations.     The solution to this system will provide the elements of M -1.   Solving Systems of Equations with The Matrix Inverse Systems of linear equations can be solved using the matrix inverse. The system must be represented by the equation                                                                                               AX=B where A  is the coefficient matrix of the system, X  is the column matrix of variables, and B  is the column matrix of constant coefficients. The solution to the system will then be A -1B  because                                                                                    Examples Explanation         The solution is (x,y)=(3,4)   Try these exercises Solve 1. Show that 2. Show that the inverse of M  is 3. Show that 4. Find the inverse of N  using a system of four equations in four unknowns. 5. Find the inverse of N  using a system of four equations in four unknowns. 6. Is the identity matrix singular or nonsingular? 7. Solve the system  using elementary row operations and matrix inverses. 8. Solve the system using elementary row orations and matrix inverses. 9. Solve the system using elementary row operations and matrix inverses. 10. Is it possible to solve a system composed of less unknowns than equations using elementary row operations and the matrix inverse? Answers to questions 1. 2.                                  3.       4. 5.                                                                                                                                    By substitution, 6. No elementary row operations are required to transform In  into In, so In-1=In. Since In has an inverse, it is nonsingular. 7. 8. 9.    . 10. Yes, but the extraneous equations must be eliminated so that the coefficient matrix is square (having the same number of equations as unknowns).

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Our basic Chinese course introduces you to the principles of ‘pinyin’ (a widely-used method of writing Mandarin Chinese using the Latin alphabet). We also help you to master the four tones and understand Chinese pronouns, numbers, greetings, telling the time, the Mandarin calendar, saying goodbye, colours, directions, nationalities and geography. The course is intended as a ‘taster’ for the rudimentary essentials of this language.  

Business Chinese

Our Business Chinese course covers different subject topics as you progress.

Level 1: Will give those with limited knowledge a foundation in basic business activities and etiquette.

Level 2:  Topics for study include, arranging company meetings and presenting product information. We will also familiarise you with the basic structure of Chinese characters.

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Level 4: You will study business situations related to placing simple orders.

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Level 8: E-commerce, supply chain management and business schemes are discussed in Level 8.

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The aims of the course are to familiarize students with the Chinese names of popular places in China, have an adequate command of the language in order to travel, conduct basic conversational techniques (e.g. asking directions, ordering food, shopping).

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Conics

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A conic section, as the name implies, is a section cut off in various ways from a circular (though not necessarily a right circular) cone by a plane. The shape of the section depends on the position of the cutting plane, that is, whether the plane cuts the cone at particular angles. Before we proceed, it is best to recapitulate the equation of a line and the condition needed for two lines to be perpendicular. A linear equation denotes a line. So, ax + by + c = 0 is the equation for a straight line.

by = -ax - c             y = -a/b x - c This is of the form y = mx + c, where m = -a/b and m = slope of the line. Suppose we have two lines

Locus

When a point moves in a plane in a manner that always satisfies a given condition, or conditions, the path it traces out is called a locus under these conditions.

Example 1:

The locus of a point that moves at a fixed distance from a fixed point is a circle.

O is the fixed point, OP is the fixed distance or radius and the point P is the locus.

Example 2:

The locus of a point equidistant from two given points is the perpendicular bisector of the line joining the two given points.

Returning to the cone, look at the following diagram.

The point V is called the vertex of the cone. O is the center of the circle formed by the base of the cone. VO is called the axis of the cone and ∠ VOA is called the semi-vertical angle. Consider a double right circular cone of semi-vertical angle (alpha), and let it be cut by a plane inclined at an angle (theta) to the axis of the cone. We will get different sections (curves) as follows:

Case I

If the plane passes through the vertex V

The curve of the intersection is a pair of straight lines passing through the vertex, and the lines are

Case II

If the plane does not pass through the vertex V

The curve of intersection is called

Hyperbola

xii. Hyperbola xiii. We may obtain the section as a pair of straight lines, a circle, a parabola, an ellipse or a hyperbola, depending on the different positions of the plane cutting the cone. These curves of intersection are called conic sections.

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