Question

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

Short answer

\( A^2 = \begin{bmatrix} 4 & 0 \\ 0 & 9 \end{bmatrix} \) and \( A^3 = \begin{bmatrix} 8 & 0 \\ 0 & 27 \end{bmatrix} \).

Step-by-step solution

Understand the Matrix

The given matrix is \( A = \begin{bmatrix} 2 & 0 \ 0 & 3 \end{bmatrix} \). Notice that it is a diagonal matrix, meaning all the non-diagonal elements are zero.

Compute \(A^2\)

To find \( A^2 \), we need to multiply the matrix \( A \) by itself: \[A^2 = \begin{bmatrix} 2 & 0 \ 0 & 3 \end{bmatrix} \begin{bmatrix} 2 & 0 \ 0 & 3 \end{bmatrix} = \begin{bmatrix} 2 \times 2 + 0 \times 0 & 2 \times 0 + 0 \times 3 \0 \times 2 + 3 \times 0 & 0 \times 0 + 3 \times 3 \end{bmatrix} = \begin{bmatrix} 4 & 0 \ 0 & 9 \end{bmatrix} .\]

Compute \(A^3\)

To find \( A^3 \), we need to multiply \( A^2 \) by \( A \):\[A^3 = A^2 \cdot A = \begin{bmatrix} 4 & 0 \ 0 & 9 \end{bmatrix} \begin{bmatrix} 2 & 0 \ 0 & 3 \end{bmatrix} = \begin{bmatrix} 4 \times 2 + 0 \times 0 & 4 \times 0 + 0 \times 3 \0 \times 2 + 9 \times 0 & 0 \times 0 + 9 \times 3 \end{bmatrix} = \begin{bmatrix} 8 & 0 \ 0 & 27 \end{bmatrix} .\]

Key concepts

Diagonal Matrix

A diagonal matrix is a special type of matrix where all the elements outside the main diagonal are zero. This means that only the elements

• in the top-left to bottom-right line (the diagonal) can have non-zero values,
• making such matrices easy to recognize and work with.

For example, in matrix \[A = \begin{bmatrix} 2 & 0 \ 0 & 3 \end{bmatrix}\]only the numbers 2 and 3 are non-zero, and they are located on the diagonal. Due to this structure, operations such as matrix multiplication become simpler. Each diagonal element behaves independently when performing exponentiation or multiplication.
This property significantly reduces the complexity in calculating powers of diagonal matrices, as you only have to raise each diagonal element to the required power separately, if the matrix is diagonal.

Exponentiation of Matrices

Exponentiation of matrices involves multiplying a matrix by itself a particular number of times, resulting in the matrix raised to that power. When dealing with diagonal matrices, exponentiation becomes an uncomplicated task. For example, finding \(A^2\) and \(A^3\) for a diagonal matrix like\[A = \begin{bmatrix} 2 & 0 \ 0 & 3 \end{bmatrix}\]is simplified because each diagonal element can be treated separately.
To find \(A^2\), you multiply the matrix by itself:\[A^2 = \begin{bmatrix} 2 & 0 \ 0 & 3 \end{bmatrix} \times \begin{bmatrix} 2 & 0 \ 0 & 3 \end{bmatrix} = \begin{bmatrix} 4 & 0 \ 0 & 9 \end{bmatrix}\].
Similarly, for \(A^3\), multiply \(A^2\) by \(A\):\[A^3 = \begin{bmatrix} 4 & 0 \ 0 & 9 \end{bmatrix} \times \begin{bmatrix} 2 & 0 \ 0 & 3 \end{bmatrix} = \begin{bmatrix} 8 & 0 \ 0 & 27 \end{bmatrix}\]. This method illustrates how diagonal matrices allow you to efficiently compute higher powers without much hassle.

Matrix Algebra

Matrix algebra is the realm of mathematics dealing with matrices and the operations involving them. This includes addition, multiplication, and exponentiation of matrices. For diagonal matrices, matrix algebra simplifies many calculations due to their structure. Consider a general concept in matrix algebra:

• When multiplying two square matrices, you usually perform a series of dot products, which can be computationally intensive.
• However, when dealing with diagonal matrices, the multiplication process is more straightforward.

This efficiency stems from certain algebraic properties of matrices: - **Associativity and Distributivity:** - Like numbers, matrices follow these useful properties, which help in organizing and simplifying calculations. - **Identity matrix and zero matrix:** - The identity matrix acts like 1 in multiplication, and the zero matrix is similar to 0 in addition. To maximize efficiency in calculations, especially for large matrices, understanding these foundational properties of matrix algebra is crucial. Diagonal matrices specifically make the application of these algebraic rules more evident and simpler, further enabling efficient calculations.