Problem 36 - Calculus for Biology and Medicine

Question

Let $$I_{3}=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]$$ Show that \(I_{3}=I_{3}^{2}=I_{3}^{3}\).

Step-by-Step Solution

Answer

The identity matrix remains unchanged when multiplied by itself: \(I_{3} = I_{3}^2 = I_{3}^3\).

Step 1: Understanding Identity Matrix

The given matrix \(I_{3}\) is a 3x3 identity matrix. An identity matrix is a square matrix with ones on the diagonal and zeros elsewhere. It serves as the multiplicative identity element in matrix multiplication, meaning any matrix multiplied by its corresponding identity matrix remains unchanged.

Step 2: Multiplying Identity Matrix by Itself Once

Calculate \(I_{3} imes I_{3}\). Recall that multiplying any matrix by the identity matrix results in the original matrix:\[I_{3} imes I_{3} = \left[\begin{array}{lll} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array}\right] \times \left[\begin{array}{lll} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array}\right] = \left[\begin{array}{lll} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array}\right]\]Thus, \(I_{3}^2 = I_{3}\).

Step 3: Multiplying Identity Matrix by Itself Twice

Now compute \(I_{3}^3\) i.e., \(I_{3} imes I_{3} imes I_{3}\). Again, multiplying by \(I_{3}\) results in the matrix remaining unchanged:\[I_{3}^3 = I_{3} imes (I_{3} imes I_{3}) = I_{3} imes I_{3} = \left[\begin{array}{lll} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array}\right]\]Thus, \(I_{3}^3 = I_{3}\).

Step 4: Conclusion

Since the identity matrix satisfies conditions for multiplicative identity, it holds that multiplying it as many times as needed \(I_{3}, I_{3}^2, I_{3}^3, ...\), results in the identity matrix itself. Therefore, \(I_{3} = I_{3}^2 = I_{3}^3\).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra, allowing us to combine two matrices to form a new one. Unlike standard arithmetic multiplication, the process for matrices involves dot products of rows and columns. Here’s a quick breakdown:

• Compatibility: To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second.
• Process: Multiply each element of a row from the first matrix by the corresponding element of a column from the second matrix, then sum the results. This gives one entry in the resulting matrix.
• Size of the result: The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix.

For an identity matrix such as \(I_{3}\), when multiplied with another 3x3 matrix, the result will always be the original matrix. This is because each row of \(I_{3}\) has a single 1 on the diagonal, effectively selecting one component from each column of the multiplied matrix.

Identity Element

In mathematics, an identity element is a special type of element in a set that leaves other elements unchanged when used in a given operation. For matrix multiplication, the identity matrix serves this purpose. Here are some key features:

• Definition: An identity matrix is a square matrix with 1's down the main diagonal and 0's elsewhere.
• Multiplicative Identity: For any matrix \(A\) of size \(n \times n\), multiplying it by the identity matrix \(I_n\) results in \(A\) itself, i.e., \(AI_n = A\) and \(I_nA = A\).
• Persistence: Multiplying an identity matrix by itself any number of times yields the same identity matrix.

In the given example, \(I_{3}\), multiplying \(I_{3}\) twice or thrice by itself still results in \(I_{3}\), demonstrating its role as the identity element.

Square Matrix

A square matrix is defined as a matrix with the same number of rows and columns. This shape is particularly important in various matrix operations, especially when dealing with concepts like determinants, eigenvalues, and more. Here's why square matrices matter:

• Uniform Dimensions: Because their row and column count is the same, operations like the determinant require the square form to be computable.
• Inversibility: Only square matrices can have inverses. If a square matrix is invertible, there exists another matrix that, when multiplied with the original, yields the identity matrix.
• Identity Matrix: The identity matrix is always a square matrix. This allows it to function effectively as the multiplicative identity in matrix operations.

In the exercise, \(I_{3}\) is a 3x3 identity matrix, illustrating both the concept of a square matrix and its role as a multiplicative identity. This dual characterization is crucial for the process of matrix multiplication.