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- Maths and statistics
- Matrices
This module discusses matrices, their order, row and column matrices, square matrices and the identity matrix. A matrix is a rectangular array of elements. Matrices are usually denoted by upper case letters. The elements are usually written within brackets. The order or shape of the matrix is determined by the...
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- Maths and statistics
- Matrices
Eigenvalues and eigenvectors are an important part of an engineer’s mathematical toolbox. They give us an understanding of how buildings, structures, automobiles and materials react in real life. Moreover they are useful for data scientists. This module does not go into each of these facets of eigenvalues and eigenvectors but...
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- Maths and statistics
- Matrices
There are rules for adding and subtracting matrices and these are reviewed in this module. Matrices of the same shape (same number of rows and columns) may be added and subtracted. Addition Matrices of the same shape (same number of rows and columns) may be added by adding the corresponding...
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- Maths and statistics
- Matrices
Matrices may be added and subtracted if they have the same shape; that is, the number of rows and columns is the same. Matrices may also be multiplied, however the requirements for multiplication are very different to that for addition/subtraction. Matrix shape or order of a matrix The shape of...
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- Maths and statistics
- Matrices
A determinant is a number that can be calculated for any square matrix. The determinant is used in calculating vector cross products, eigenvalues, eigenvectors and solving simultaneous equations. Determinant of a \(2\times2\) matrix The determinant of a \(2\times2\) matrix is called a second order determinant. Let \[\begin{align*} A & =\left[\begin{array}{cc}...
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- Maths and statistics
- Matrices
It is helpful to understand the definition of a number of different types of “special” matrices. Transpose of a matrix The transpose of a matrix \(\mathbf{A}\) is denoted \(\mathbf{A}^{T}\)and is found by interchanging the rows and the columns. The first row becomes the first column, the second row becomes the...
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- Maths and statistics
- Matrices
Matrices can be used to solve systems of equations by using elementary row operations and the augmented matrix. The augmented matrix Consider the following system of equations: \[\begin{align*} x+2y-z & =-3\\ 2x-3y+2z & =13\\ -x+5y-4z & =-19 \end{align*}\] The corresponding augmented matrix for this system is \[\begin{align*} \left[\begin{array}{ccc} 1 &...
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- Maths and statistics
- Matrices
For any system of equations, there may be: Infinitely many solutions No solution A unique solution Coefficient matrix Consider the following system of equations: \[\begin{align*} 2x+4y-z & =9\\ x-y+2z & =-4\\ -x+y-z & =3 \end{align*}\] These may be written in the matrix form: \[\begin{align*} \left[\begin{array}{ccc} 2 & 4 & -1\\...
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- Maths and statistics
- Matrices
In matrix algebra, we can add, subtract and multiply matrices subject to conditions on the matrix shape (or order). While matrix algebra does not have a division operation, there is multiplication by the inverse matrix. Definition Let \(I\) denote the identity matrix. That is the matrix containing ones on the...
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- Maths and statistics
- Matrices
Understanding the inverse of a 3x3 matrix equips you with skills for solving systems of linear equations, crucial in fields like engineering, physics, and computer science. This knowledge simplifies complex problems and enhances your ability to perform matrix operations efficiently. Mastering this concept will boost your mathematical proficiency and problem-solving...
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Matrices are used in a wide range of fields and applications to represent and manipulate data in a structured way. The links on this page lead to useful resources for students who are starting out with matrices and also those who would like to add to their existing knowledge. M1...
