Problem 31 - Algebra and Trigonometry

Question

The matrices \(A, B, C, D, E, F, G\) and \(H\) are defined as follows. $$ A=\left[\begin{array}{rr}{2} & {-5} \\ {0} & {7}\end{array}\right] \quad B=\left[\begin{array}{rrr}{3} & {\frac{1}{2}} & {5} \\ {1} & {-1} & {3}\end{array}\right] \quad C=\left[\begin{array}{rrr}{2} & {-\frac{5}{2}} & {0} \\ {0} & {2} & {-3}\end{array}\right] $$ $$ D=\left[\begin{array}{ll}{7} & {3}\end{array}\right] \quad E=\left[\begin{array}{l}{1} \\ {2} \\ {0}\end{array}\right] \quad F=\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right] $$ $$ G=\left[\begin{array}{rrr}{5} & {-3} & {10} \\ {6} & {1} & {0} \\ {-5} & {2} & {2}\end{array}\right] \quad H=\left[\begin{array}{rr}{3} & {1} \\ {2} & {-1}\end{array}\right] $$ Carry out the indicated algebraic operation, or explain why it cannot be performed. $$ A^{2} \quad \text { (b) } A^{3} $$

Step-by-Step Solution

Answer

\( A^2 = \begin{bmatrix} 4 & -45 \\ 0 & 49 \end{bmatrix} \), \( A^3 = \begin{bmatrix} 8 & -335 \\ 0 & 343 \end{bmatrix} \).

Step 1: Check Matrix Compatibility

Matrix \(A\) is a 2x2 matrix. To perform matrix multiplication such as finding \(A^2\) or \(A^3\), this operation is possible because the number of columns in matrix \(A\) (2) matches the number of rows in matrix \(A\) (2). Thus, \(A\) can be multiplied by itself.

Step 2: Calculate \( A^2 \)

Calculate \( A^2 \) by multiplying matrix \( A \) by itself: \[A^2 = A \times A = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} \times \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix}\]Compute individual elements:- First row, first column: \(2 \times 2 + (-5) \times 0 = 4\)- First row, second column: \(2 \times (-5) + (-5) \times 7 = -10 - 35 = -45\)- Second row, first column: \(0 \times 2 + 7 \times 0 = 0 \)- Second row, second column: \(0 \times (-5) + 7 \times 7 = 49\)Thus, \[ A^2 = \begin{bmatrix} 4 & -45 \ 0 & 49 \end{bmatrix} \]

Step 3: Calculate \( A^3 \)

Calculate \( A^3 \) by multiplying \( A^2 \) by \( A \): \[A^3 = A^2 \times A = \begin{bmatrix} 4 & -45 \ 0 & 49 \end{bmatrix} \times \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix}\]Compute individual elements:- First row, first column: \(4 \times 2 + (-45) \times 0 = 8\)- First row, second column: \(4 \times (-5) + (-45) \times 7 = -20 - 315 = -335\)- Second row, first column: \(0 \times 2 + 49 \times 0 = 0\)- Second row, second column: \(0 \times (-5) + 49 \times 7 = 343\)Thus, \[ A^3 = \begin{bmatrix} 8 & -335 \ 0 & 343 \end{bmatrix} \]

Key Concepts

2x2 matrices

A 2x2 matrix is a rectangular array composed of 2 rows and 2 columns. Each element or entry in a 2x2 matrix is typically denoted by the variable letters, such as the elements of matrix \( A = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} \). Understanding how these matrices operate is crucial for performing further operations such as matrix multiplication.

The organization of the elements in a 2x2 matrix heavily influences how operations like addition, subtraction, and particularly multiplication are carried out. When looking at matrix multiplication within 2x2 matrices, the number of columns in the first matrix must match the number of rows in the second. This condition is inherently met when a square matrix (like a 2x2) multiplies another 2x2. This characteristic makes 2x2 matrices simpler yet fundamental building blocks in more complex matrix algebra.

algebraic operations

In matrix algebra, operations such as addition and multiplication take on a more structured form than simple arithmetic. With 2x2 matrices, these operations need particular conditions for them to be valid.

When performing matrix multiplication, each element of the resulting matrix is calculated by taking the dot product of rows and columns. Let's see how this works with matrix \( A \):

• First row, first column: Calculated as \( 2 \times 2 + (-5) \times 0 = 4 \).
• First row, second column: Calculated as \( 2 \times (-5) + (-5) \times 7 = -45 \).

This continues for each combination, ensuring that every element is the sum of products, capturing the interaction between rows and columns.

Matrix multiplication is unlike regular multiplication, as it is not commutative; that is, \( A \times B eq B \times A \). This nature of multiplication makes understanding compatibility and arrangement essential in solving matrix operations.

matrix powers

Matrix powers involve raising a matrix to an integer power, similar to raising numbers. When you see \( A^n \), it denotes multiplying the matrix \( A \) by itself \( n \) times. For a 2x2 matrix like \( A \), you start by finding \( A^2 = A \times A \) and use that result to find \( A^3 = A^2 \times A \).

For instance, in our case:
1. First, we calculated \( A^2 \) by multiplying \( A \) itself:

• The product gave us the matrix \( \begin{bmatrix} 4 & -45 \ 0 & 49 \end{bmatrix} \).

2. Then, to find \( A^3 \), we multiply this result by \( A \) again.
Within each of these computations, the elements are methodically determined by dot products across respective rows and columns from the matrices involved.

Understanding how matrix powers work is key for advancing in matrix applications, as they can represent transformations, rotations, or other changes repeatedly applied.