Question

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

Short answer

\(A^2\) is the identity matrix, and \(A^3 = A\).

Step-by-step solution

Understanding the Problem

We are given a matrix \( A \) and asked to find its square \( A^2 \) and cube \( A^3 \). Matrix \( A \) is defined as \( A = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \). To solve this, we will perform matrix multiplication.

Calculating \( A^2 \)

Compute \( A^2 = A \times A \):\[A = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \quad \text{and}\quad A = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}\]Multiply the matrices:\[A^2 = \begin{bmatrix} (0 \cdot 0 + 1 \cdot 1) & (0 \cdot 1 + 1 \cdot 0) \ (1 \cdot 0 + 0 \cdot 1) & (1 \cdot 1 + 0 \cdot 0) \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\]Thus, \( A^2 = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \).

Calculating \( A^3 \)

Now find \( A^3 = A^2 \times A \):\[A^2 = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}, \quad A = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}\]Multiply to find \( A^3 \):\[A^3 = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \times \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} = \begin{bmatrix} (1 \cdot 0 + 0 \cdot 1) & (1 \cdot 1 + 0 \cdot 0) \ (0 \cdot 0 + 1 \cdot 1) & (0 \cdot 1 + 1 \cdot 0) \end{bmatrix} = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}\]Therefore, \( A^3 = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \).

Conclusion of Calculations

We determined that:\[ A^2 = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \] and \[ A^3 = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \]. These results reflect the fact that \( A^3 = A \) which implies \( A \) is an involutory matrix (its square is the identity matrix).

Key concepts

Matrix Multiplication

Matrix multiplication involves taking two matrices and producing a new matrix by performing a series of calculations. When multiplying two matrices, such as matrices \(A\) and \(B\), what happens is you take each element from a row of the first matrix and multiply it by each corresponding element in a column of the second matrix. Then, you sum those results. This is why multiplying matrices can sound complicated, but it follows a straightforward pattern:

• For each row in the first matrix and each column in the second matrix, multiply the corresponding entries and sum them.
• Repeat for all row and column pairs.
• The resulting matrix, or 'product', has dimensions based on the row count of the first and the column count of the second matrices.

In our exercise, multiplying matrix \(A\) by itself (\(A \times A\)) gave us the identity matrix, demonstrating how these steps apply practically.

Involutory Matrix

An involutory matrix is a specific kind of square matrix. It has a unique property: when the matrix is multiplied by itself, the result is the identity matrix. Essentially, if matrix \(A\) is the original matrix, and \(A^2 = I\), where \(I\) is the identity matrix, then \(A\) is involutory.

• This special property means that the involutory matrix "undoes" itself when squared.
• It's like pressing a reset button: the transformation applied by the matrix can be reversed by applying the matrix again.
• This reversibility is particularly useful in linear algebra and computational applications, providing a mathematical way to quickly revert systems to their initial states.

In our case, the matrix \(A = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}\) is involutory because it swaps its entries to yield an identity matrix upon squaring.

Identity Matrix

The identity matrix is like the number 1 in ordinary multiplication. When you multiply any matrix by the identity matrix, the original matrix remains unaffected. This matrix is a square matrix with all the main diagonal elements as 1 and all off-diagonal elements as 0, such as
\[ I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \]

• The identity matrix acts as a neutral element in matrix multiplication.
• For any matrix \(A\), \(A \times I = A\) or \(I \times A = A\).
• It's crucial in many areas of mathematics and computer science because it permits the establishment of matrix inverses and other properties.

In our problem, when matrix \(A\) was squared, it produced the identity matrix, showing that it can "reset" a transformation, thus illustrating why \(A\) is called an involutory matrix.