unit matrix
a square matrix where the leading diagonal entries are 1.
A unit matrix is a square matrix in which all the elements of the diagonal are equal to 1, and all the other elements are equal to 0. It is also known as an identity matrix. The standard notation for a unit matrix is I_n, where n represents the dimensions of the matrix. For example, a unit matrix of dimension 3×3 would be represented as:
I_3 = 1 0 0
0 1 0
0 0 1
The main property of a unit matrix is that when multiplied by any matrix A, the resulting matrix is always equal to A itself. This property is known as the multiplicative identity property and is very useful in many mathematical and scientific applications. Additionally, the unit matrix is also inversible, meaning that its inverse matrix is equal to itself.
Other important properties of unit matrices include:
– The determinant of a unit matrix is always equal to 1.
– The trace of a unit matrix is equal to the number of dimensions.
– The unit matrix is always a symmetric matrix.
– The unit matrix is orthogonal.
Overall, the unit matrix is a fundamental concept in linear algebra and is used in a wide range of mathematical and scientific applications.
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