Question

Let \(A=\left[\begin{array}{ccc}-1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5\end{array}\right] .\) Find \(A^{2}\) and \(A^{3} .\)

Short answer

\( A^2 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 9 & 0 \\ 0 & 0 & 25 \end{bmatrix} \), \( A^3 = \begin{bmatrix} -1 & 0 & 0 \\ 0 & 27 & 0 \\ 0 & 0 & 125 \end{bmatrix} \).

Step-by-step solution

Identify the Type of Matrix

The matrix \( A \) is a diagonal matrix, which means all of its non-zero elements are located along the diagonal from the top left to the bottom right. The elements outside this diagonal are zero.

Understanding Diagonal Matrix Multiplication

For a diagonal matrix \( A \), the square \( A^2 \) is calculated by squaring each element on the diagonal. Similarly, the cube \( A^3 \) is found by cubing each diagonal element.

Calculate \( A^2 \)

To find \( A^2 \):- Square each diagonal element.- The resulting matrix is \( A^2 = \begin{bmatrix} (-1)^2 & 0 & 0 \ 0 & 3^2 & 0 \ 0 & 0 & 5^2 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \ 0 & 9 & 0 \ 0 & 0 & 25 \end{bmatrix} \).

Calculate \( A^3 \)

To find \( A^3 \):- Cube each diagonal element.- The resulting matrix is \( A^3 = \begin{bmatrix} (-1)^3 & 0 & 0 \ 0 & 3^3 & 0 \ 0 & 0 & 5^3 \end{bmatrix} = \begin{bmatrix} -1 & 0 & 0 \ 0 & 27 & 0 \ 0 & 0 & 125 \end{bmatrix} \).

Summarize the Result

The matrices found are \( A^2 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 9 & 0 \ 0 & 0 & 25 \end{bmatrix} \) and \( A^3 = \begin{bmatrix} -1 & 0 & 0 \ 0 & 27 & 0 \ 0 & 0 & 125 \end{bmatrix} \).

Key concepts

Matrix Multiplication

Matrix multiplication involves combining two matrices to produce a new matrix. It is a fundamental operation in linear algebra, integral to various applications, such as computer graphics and solving systems of equations. The process is quite different from simply multiplying numbers. You need to follow a specific set of rules:

• Ensure the two matrices are compatible for multiplication: The number of columns in the first matrix must equal the number of rows in the second matrix.
• Multiply the elements of each row in the first matrix by the corresponding elements of each column in the second matrix, summing the products.
• The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.

Practically, you don't just multiply element by element; you multiply across rows and down columns, accumulating the products. When considering a diagonal matrix like the one in the exercise, this process simplifies. You only need to square or cube each diagonal element because the elements outside the main diagonal are already zero. This property makes working with diagonal matrices much simpler.

Matrix Exponentiation

Matrix exponentiation is the process of raising a matrix to a power. This can be thought of as performing repeated matrix multiplication of a matrix by itself. This concept has practical applications, such as computing powers of transition matrices in Markov chains or solving certain types of differential equations.In the context of the original exercise, where matrix \( A \) is a diagonal matrix, matrix exponentiation is straightforward. To compute \( A^2 \) or \( A^3 \):

• Directly raise each diagonal element to the given power.
• Maintain zeros in all off-diagonal positions because they do not change regardless of the power.

So, for the diagonal matrix \( A = \begin{bmatrix} -1 & 0 & 0 \ 0 & 3 & 0 \ 0 & 0 & 5 \end{bmatrix} \), \( A^2 \) and \( A^3 \) are easily calculated as:

• \( A^2 = \begin{bmatrix} (-1)^2 & 0 & 0 \ 0 & 3^2 & 0 \ 0 & 0 & 5^2 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \ 0 & 9 & 0 \ 0 & 0 & 25 \end{bmatrix} \)
• \( A^3 = \begin{bmatrix} (-1)^3 & 0 & 0 \ 0 & 3^3 & 0 \ 0 & 0 & 5^3 \end{bmatrix} = \begin{bmatrix} -1 & 0 & 0 \ 0 & 27 & 0 \ 0 & 0 & 125 \end{bmatrix} \)

Matrix exponentiation in diagonal matrices benefits from this simplicity, avoiding complex calculations typically required for non-diagonal matrices.

Diagonal Elements

Understanding diagonal elements is crucial, especially when dealing with diagonally dominant matrices or diagonal matrices. The diagonal elements of a matrix are those situated on the line that runs from the top left to the bottom right of the matrix. These elements are significant because, in diagonal matrices, all non-diagonal elements are zero.Here's why diagonal matrices are interesting:

• Their determinant is simply the product of their diagonal elements.
• For matrix exponentiation, each diagonal element can be raised to a power independently, which simplifies the calculations immensely.
• They simplify many mathematical operations, making them computationally efficient, especially for larger matrices.

In the specific case of a matrix \( A = \begin{bmatrix} -1 & 0 & 0 \ 0 & 3 & 0 \ 0 & 0 & 5 \end{bmatrix} \), the diagonal elements are \(-1, 3,\) and \(5\). These elements can entirely determine operations like calculating powers of the matrix, making the process much more manageable. This is why understanding diagonal elements is key to appreciating the efficiency and beauty of working with diagonal matrices.