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A detailed tutorial on the identity matrix. Step by step tutorial including several examples of the identity matrix and how to solve it for reference.

An indentity matrix is a matrix that is said to be of size n. It is considered to be the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. The identity matrix is denoted as the variable I. The identity matrix has some extremely important properties of its own, especially multiplication properties. It is a unique type of matrix that is found rarely, but is used very often in several different branches of math.

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A detailed tutorial on what a scalar is. Step by step tutorial including several examples of scalars and how they relate to vectors for reference.

A scalar is a number that relates vectors on a vector space through the process of scalar multiplication. A scalar can be taken from any set of numbers, including rational, algebraic, real, and complex sets of numbers. The scalar is always a real number. A scalar is a single component, and things such as vectors, matrices, and tensors can be reduced to a scalar.

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A detailed tutorial on the main diagonal of a matrix. Step by step tutorial including several examples of main diagonals for reference.

The main diagonal of a matrix is the diagonal that starts at the top left corner, and continues down and to the right one step until either the other corner is reached (square matrices only), the bottom of the matrix is reached, or the right side of the matrix is reached. The main diagonal is also sometimes called the primary diagonal or the leading diagonal

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A detailed tutorial on upper and lower triangular matrices. Step by step tutorial including several examples of triangular matrices for reference.

A triangular matrix is a kind of square matrix where an element above or below the main diagonal is 0. This gives the true elements of the matrix a triangle shape, which is how it got its name. An upper triangular matrix is sometimes called a right triangular matrix. The matrix is up in the right upper corner, and the 0 element is in the lower left corner. A lower triangular matrix is sometimes called a left triangular matrix. The matrix is in the left bottom corner, and the 0 element is in the upper right corner.

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A detailed tutorial on the transpose of a matrix. Step by step tutorial including several examples of the transpose of a matrix for reference.

When you transpose a matrix, it is simply a way of saying that you write the matrix in a different way – this creates a new matrix. There are three ways you can transpose a matrix. The first way is to write the rows of your matrix as columns instead. The second way is to write the columns of your matrix as rows instead. And the third way is to reflect your matrix by its main diagonal. All of these actions accomplish the same thing, so it does not matter which method you use. When people talk about transposing something, they are usually referring to matrices.

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A detailed tutorial on eigenvalues and eigenvectors. Step by step tutorial including several examples of eigenvalues and eigenvectors for reference.

Eigenvalues and eigenvectors are related concepts commonly used in linear algebra. More specifically, they are properties of a matrix. They give very important information about a matrix, and are used in matrix factorization. Assuming that a matrix is a diagonal matrix (a square matrix or a similar matrix that you can calculate diagonals on), then the eigenvalues are the numbers on the diagonal and the eigenvectors are the basis vectors to which there numbers refer. You cannot have an eigenvector without an eigenvalue. However, you can have an eigenvalue without an eigenvector.

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A detailed tutorial on square matrices and how to identify them. Step by step tutorial including several examples of square matrices for reference.

A square matrix is a simple matrix in the shape of a square. It has the same number of rows and columns. Square matrices are called nxn matrces. The most common values for n are 2 and 3. Two columns and rows is the smallest amount of rows and columns a square matrix can have – matrices with only one value are not considered to be square.

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A detailed tutorial on find the trace of a matrix. Step by step tutorial including several examples of how to find the trace for reference.

The trace of a square matrix is defined to be the sum of the elements on the main diagonal of the matrix. This can be mathematically expressed as:

Remember, the trace is only defined for square matrices – not any other kind of matrix.

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A detailed tutorial on how to find the determinant. Step by step tutorial including several examples of finding the determinant for reference.

The determinant is a number that is associated with a square matrix. In a mathematical sense, the determinant is a scale factor for measure when the matrix is regarded as a linear transformation. The determinant is denoted by two bars on either side of the matrix, which can be confused with the absolute value of the matrix. The determinant is found by subtracting the products of the diagonals of the matrix, at least in a 2×2 matrix.

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A detailed tutorial on the Rule of Sarrus. Step by step tutorial including several examples of the Rule of Sarrus and determinants for reference.

The Rule of Sarrus is a method used to compute the determinant of a 3×3 matrix. Mathematically stated, if you are given a 3×3 matrix, you can compute the determinant by repeating the first two columns of the matrix behind the third column, so that you have 5 columns in a row. This forms a 3×5 matrix. Then you add the products of the diagonals going from top to bottom (left to right), and subtract the products going from bottom to top (left to right). This can also be used for 2×2 matrices, but the rule used is a little different.