Question

Show that det \(I=1\) for any identity matrix \(l .\)

Short answer

The determinant of an identity matrix, \(det(I)\), is 1 for any size.

Step-by-step solution

Understand the Identity Matrix

An identity matrix, denoted by \(I\), is a square matrix with all diagonal elements equal to 1 and all other elements equal to 0. The size of the matrix, \(n \times n\), determines the dimensions but the structure remains the same for all sizes.

Determine the Property of the Identity Matrix

The identity matrix serves as the multiplicative identity in matrix algebra, meaning that when any matrix \(A\) is multiplied by \(I\), the resulting product is \(A\) itself.

Calculate the Determinant of a 2x2 Identity Matrix

For a 2x2 identity matrix, \(I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\), the determinant is calculated as \(det(I) = (1)(1) - (0)(0) = 1\).

Generalize to an nxn Identity Matrix

For a generic \(n \times n\) identity matrix, the formula for the determinant is the product of its diagonal elements. Since all diagonal elements of the identity matrix # are 1, the determinant is the product of \(n\) ones, which is 1, i.e., \(det(I) = 1^n = 1\).

Conclusion

The determinant of any identity matrix, regardless of its size, is always 1 because it does not alter any matrix it is multiplied with and its diagonal elements are all 1.

Key concepts

Identity Matrix

An identity matrix is a special kind of matrix important in mathematics, especially in matrix algebra. It is called an "identity" matrix because it acts like the number 1 in multiplication for any matrix. Imagine multiplying any number by 1, it does not change the number; similarly, multiplying a matrix by an identity matrix leaves the original matrix unchanged. This property makes the identity matrix fundamental in the study of matrices.

The identity matrix is a square matrix, which means it has the same number of rows and columns, denoted as an \(n \times n\) matrix. Every element on the main diagonal (from the top left to the bottom right) of the identity matrix is 1, and all other elements are 0. For example, a 2x2 identity matrix looks like this:
\[I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}. \]
For larger identity matrices, the same pattern continues; only the diagonal elements are 1, and everything else is 0.

• 2x2 Identity Matrix:: \(\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\)
• 3x3 Identity Matrix:: \(\begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\)

The identity matrix has versatile applications in computational mathematics and is fundamental to understanding more complex matrix manipulations.

Matrix Algebra

Matrix algebra is the part of mathematics dealing with matrices that involves the operations of addition, subtraction, multiplication, and finding determinants, among others. Matrices are useful for solving various mathematical problems, including systems of linear equations and transformations in space.

One of the most crucial operations in matrix algebra is matrix multiplication. Unlike number multiplication, where the order doesn't matter (i.e., 2 * 3 is the same as 3 * 2), the order in which matrices are multiplied can affect the result. In matrix algebra, the identity matrix serves as the neutral element for multiplication.

• When you multiply any matrix \(A\) by the identity matrix \(I\), the product is \(A\) itself, written as \(A \cdot I = I \cdot A = A\).
• This property resembles multiplying a number by 1 and is extremely useful when solving matrix equations or simplifying matrix expressions.

Furthermore, understanding matrix operations helps simplify many algebraic computations, enabling the use of matrices in computer graphics, quantum mechanics, and applied mathematics.

Determinant Properties

The determinant is a scalar value that can be computed from a square matrix, providing insight into certain properties of the matrix. For example, it can tell us if a matrix has an inverse or if certain systems of equations have unique solutions. The determinant can also give us a way of using matrix data in transformations and solving equations.

One important property of the determinant is its behavior with the identity matrix. Calculating the determinant of an identity matrix is straightforward, because it is always 1, regardless of the size of the matrix. Here's why:

• The determinant of a matrix is the product of its diagonal elements.
• In an identity matrix, all diagonal elements are 1, so the product is 1.
• Therefore, \(\text{det}(I) = 1^n = 1\) for an \(n \times n\) identity matrix.

This does not alter when you multiply a matrix by an identity matrix, as the determinant of the product remains based on the original matrix. The properties of determinants are widely used in advanced topics of mathematics, helping provide further insights into matrix operations and their effects.