Problem 99 - Precalculus

Question

$$\text { For } A=\left[\begin{array}{ll}a_{11} & a_{12} \\\a_{21} & a_{22}\end{array}\right], \text { find } A^{2}$$

Step-by-Step Solution

Answer

\(A^2 = \left[\begin{array}{cc} a_{11}^2 + a_{12}a_{21} & a_{11}a_{12} + a_{12}a_{22} \\ a_{21}a_{11} + a_{22}a_{21} & a_{21}a_{12} + a_{22}^2 \end{array}\right]\)

Step 1: Understand Matrix Multiplication

Matrix multiplication requires the sum of the products of the rows of the first matrix and the columns of the second matrix. For a matrix \(A\), the multiplication \(A \times A\) results in a matrix where each entry \(c_{ij}\) is calculated as \(c_{ij} = a_{i1}a_{1j} + a_{i2}a_{2j}\).

Step 2: Multiply the First Row by First Column

Calculate the element in the first row, first column of \(A^2\):\[c_{11} = a_{11}a_{11} + a_{12}a_{21}\]Which simplifies to:\[c_{11} = a_{11}^2 + a_{12}a_{21}\]

Step 3: Multiply the First Row by Second Column

Calculate the element in the first row, second column of \(A^2\):\[c_{12} = a_{11}a_{12} + a_{12}a_{22}\]Which simplifies to:\[c_{12} = a_{11}a_{12} + a_{12}a_{22}\]

Step 4: Multiply the Second Row by First Column

Calculate the element in the second row, first column of \(A^2\):\[c_{21} = a_{21}a_{11} + a_{22}a_{21}\]Which simplifies to:\[c_{21} = a_{21}a_{11} + a_{22}a_{21}\]

Step 5: Multiply the Second Row by Second Column

Calculate the element in the second row, second column of \(A^2\):\[c_{22} = a_{21}a_{12} + a_{22}a_{22}\]Which simplifies to:\[c_{22} = a_{21}a_{12} + a_{22}^2\]

Step 6: Combine Results Into A Matrix

Combine all calculated elements to represent matrix \(A^2\):\[A^2 = \left[\begin{array}{cc} a_{11}^2 + a_{12}a_{21} & a_{11}a_{12} + a_{12}a_{22} \ a_{21}a_{11} + a_{22}a_{21} & a_{21}a_{12} + a_{22}^2 \end{array}\right]\]

Key Concepts

Understanding Matrix Powers

Matrix powers involve repeatedly multiplying a matrix by itself. It connects to the broader topic of matrix multiplication. For a square matrix, say matrix \(A\), finding the power \(n\) means multiplying \(A\) by itself \(n\) times. If we consider \(A^2\), this simply means performing the operation \(A \times A\).
\(A^2\) helps find insights into linear transformations and can simplify complex equations in fields like physics or computer science.
- Identical size: Both matrices in multiplication must have the same dimensions for matrix powers.- Exponential growth: As powers increase, the original transformation effect becomes more pronounced.
Understanding matrix powers can greatly help in fields like dynamical systems or Markov chains, where they are used to analyze steady states or long-term predictions.

Exploring Matrix Algebra

Matrix algebra is the set of tools and operations that tell us how to manipulate matrices. It includes addition, subtraction, and importantly, multiplication of matrices.
One essential property is the order of matrices. The multiplication order is critical because it is not commutative; \(AB eq BA\) in most cases. Hence, while working with matrices, it is crucial to ensure that matrices being multiplied have compatible dimensions.
Key points:

• Associative property: \(A(BC) = (AB)C\)
• Distributive property: \(A(B + C) = AB + AC\)
• Identity matrix: Acts like the number 1 in multiplication of matrices, \(IA = AI = A\)

Understanding these properties helps simplify calculations and solve matrix equations that appear in areas like economics, engineering, and computer graphics.

Performing Matrix Operations

Matrix operations are actions such as addition, subtraction, and especially multiplication, which is our focus here.
Typically, when multiplying two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. The product, say matrix \(C\), will have dimensions derived from the outer dimensions of the factor matrices.
- In our exercise: We calculated elements of \(A^2\) by systematically multiplying rows of the original matrix by its columns.- The method involves multiplying each element of the row by its corresponding column and summing these products to obtain single elements.
This step-by-step method reveals how even simple matrices can convey complex transformations and is core to problems spanning computer algorithms, physics simulations, and economics models.