Problem 49 - Algebra for College Students

Question

If \(A=\left[\begin{array}{rr}1 & -1 \\ 2 & 3\end{array}\right]\), calculate \(A^{2}\) and \(A^{3}\)

Step-by-Step Solution

Answer

\(A^2 = \begin{bmatrix} -1 & -4 \\ 8 & 7 \end{bmatrix}\), \(A^3 = \begin{bmatrix} -9 & -11 \\ 22 & 13 \end{bmatrix}\)

Step 1: Compute A²

\(A^2 = A \cdot A = \begin{bmatrix} 1&-1\\2&3 \end{bmatrix}\begin{bmatrix} 1&-1\\2&3 \end{bmatrix}\)

\(= \begin{bmatrix} 1-2 & -1-3 \\ 2+6 & -2+9 \end{bmatrix} = \begin{bmatrix} -1 & -4 \\ 8 & 7 \end{bmatrix}\)

Step 2: Compute A³

\(A^3 = A^2 \cdot A = \begin{bmatrix} -1&-4\\8&7 \end{bmatrix}\begin{bmatrix} 1&-1\\2&3 \end{bmatrix}\)

\(= \begin{bmatrix} -1-8 & 1-12 \\ 8+14 & -8+21 \end{bmatrix} = \begin{bmatrix} -9 & -11 \\ 22 & 13 \end{bmatrix}\)

Key Concepts

Matrix Powers

When we talk about matrix powers, we are referring to the process of multiplying a matrix by itself a certain number of times. In this context, matrix powers are quite similar to calculating powers of numbers, but here, we deal with matrices instead. For example, to calculate \(A^2\), we multiply the matrix \(A\) by itself:

• Start with the initial matrix \(A\) as given.
• Multiply matrix \(A\) by itself to form \(A^2\).

This process is recursive. To find a higher power like \(A^3\), you simply multiply the result of \(A^2\) by the matrix \(A\) again. It becomes clearer:

• Calculate \(A^2 = A \times A\).
• Then, calculate \(A^3 = A^2 \times A\).

Each time you multiply matrices, follow the rules of matrix multiplication. This is done by taking each element of the relevant rows and columns, and then adding up the products.

Square Matrices

A square matrix is a matrix that has the same number of rows and columns. The matrix \(A\) provided in the exercise is a square matrix because it has 2 rows and 2 columns:

• This specific example is a 2x2 matrix.
• Square matrices are significant in linear algebra as their properties facilitate various operations, including taking powers.

One special property of square matrices is their potential to be invertible. For square matrices, it is also possible to define determinants and eigenvalues. Please note that these additional properties are useful beyond simple multiplication.Since the given matrix \(A\) is square, calculating its powers like \(A^2\) or \(A^3\) is straightforward and follows typical arithmetic used in matrices. The uniform shape allows repeated multiplication without the need for conformability checks, as is required with non-square matrices.

Matrix Operations

Matrix operations encompass a variety of calculations you can perform on matrices, such as addition, subtraction, and multiplication. In our case, for solving the exercise, we focus on multiplication. Here’s a quick breakdown:

• **Addition/Subtraction:** Just add or subtract corresponding elements from each matrix.
• **Multiplication:** Follow the rule of multiplying rows by columns and summing them up.

Matrix multiplication is not only about operations but also has rules that need to be followed such as:

• The number of columns in the first matrix must equal the number of rows in the second.
• The order in multiplication matters; \(AB eq BA\) in most cases.

Applying these rules, you multiply matrix \(A\) by itself to get \(A^2\), and then \(A^2\) by \(A\) again for \(A^3\). This highlights the importance of carefully applying these operations and understanding their rules.