Question

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

Short answer

\(A^2 = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}\), \(A^3 = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\).

Step-by-step solution

Understand the Problem

To solve this problem, we need to calculate the square and the cube of the given matrix \(A\). The matrix \(A\) is a \(3\times3\) matrix, and we need to find \(A^2 = A \cdot A\) and \(A^3 = A \cdot A \cdot A\).

Calculate \(A^2\)

The matrix \(A\) is \[A = \begin{bmatrix} 0 & 0 & 1 \ 0 & 0 & 0 \ 0 & 1 & 0 \end{bmatrix}\]To find \(A^2\), multiply \(A\) by itself:\[A^2 = \begin{bmatrix} 0 & 0 & 1 \ 0 & 0 & 0 \ 0 & 1 & 0 \end{bmatrix}\begin{bmatrix} 0 & 0 & 1 \ 0 & 0 & 0 \ 0 & 1 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 1 & 0 \ 0 & 0 & 0 \ 0 & 0 & 1 \end{bmatrix}\]

Calculate \(A^3\)

To get \(A^3\), multiply \(A\) by \(A^2\):\[A^3 = A \cdot A^2 = \begin{bmatrix} 0 & 0 & 1 \ 0 & 0 & 0 \ 0 & 1 & 0 \end{bmatrix}\begin{bmatrix} 0 & 1 & 0 \ 0 & 0 & 0 \ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 1 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{bmatrix}\]

Verify the Calculations

Review each matrix multiplication step to ensure the calculations were done correctly. For instance, in \(A^2\), verify each element of the resulting matrix by cross-checking the row-by-column multiplication with the original matrix \(A\). Repeating this process ensures accuracy before compiling the final results for \(A^2\) and \(A^3\).

Key concepts

Matrix Powers

Matrix powers refer to multiplying a matrix by itself multiple times. In the exercise above, we need to find the second and third powers of matrix \(A\), which involves matrix multiplication. Matrix powers are used in many areas like computer graphics and solving linear differential equations, making them valuable to understand. When raising a matrix to a power, it helps to first ensure the matrix is square, meaning it has the same number of rows and columns. Here, since the given matrix \(A\) is a \(3\times3\) matrix, you can compute powers like \(A^2\) (\(A\) times \(A\)) or \(A^3\) (\(A^2\) times \(A\)) quite straightforwardly. An important thing to remember with matrix powers is that unlike numbers, the results can vary drastically. For instance, repetitive multiplication can often lead matrices to become zero matrices or identity matrices under certain conditions. Therefore, checking calculations at each step like in the example ensures your understanding and accuracy.

Matrix Operations

Matrix operations involve tasks such as addition, subtraction, and importantly for this exercise - multiplication. Understanding these operations is crucial when working with matrices of any size.Here, multiplication is the key focus. When multiplying matrices, remember each element in the product results from adding the products of the rows of the first matrix with the columns of the second. This row-by-column method emphasizes matching the dimensions; a matrix \(A\) with dimensions \(m \times n\) can only be multiplied by a matrix \(B\) with dimensions \(n \times p\). In our exercise with matrix \(A\), since it's \(3 \times 3\), it’s able to multiply with itself. Each step of the multiplication must be individually verified. For instance, to calculate the element in the first row, first column of the resulting matrix from \(A^2\), you multiply each component of the first row of the first matrix by the corresponding component in the first column of the second matrix, and sum the results.

3x3 Matrices

A \(3 \times 3\) matrix consists of three rows and three columns, resulting in a matrix composed of 9 elements. These matrices often represent transformations in three-dimensional space and are very useful in computations across engineering, physics, and computer science fields.In this exercise, the matrix \(A\) was given as:\[A = \begin{bmatrix} 0 & 0 & 1 \ 0 & 0 & 0 \ 0 & 1 & 0 \end{bmatrix}\]Working with \(3\times3\) matrices may seem complex at first, but the operations are explainable step-by-step. Understanding the layout and position of each entry in the matrix helps immensely when performing operations like matrix multiplication.One practicality in the context of this exercise is that \(3\times3\) matrices are small enough to compute manually yet large enough to illustrate more complex vector transformations — serving as a great learning tool. Moreover, insights gained from understanding these operations extend to larger matrices often used in data science and machine learning.