Problem 31 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

(a) \(A^2 = \begin{bmatrix} 4 & -45 \\ 0 & 49 \end{bmatrix}\); (b) \(A^3 = \begin{bmatrix} 8 & -385 \\ 0 & 343 \end{bmatrix}\)

Step 1: Determine A^2 Existence

For a square matrix A, the square matrix A^2 exists if A is 2x2. Check given matrix A: \[ A = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} \]. A is a 2x2 matrix, so A^2 can be calculated.

Step 2: Calculate A^2

To find \( A^2 \), multiply matrix A by itself:\[A^2 = A \times A = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} \times \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} = \begin{bmatrix} 2 \cdot 2 + (-5) \cdot 0 & 2 \cdot (-5) + (-5) \cdot 7 \ 0 \cdot 2 + 7 \cdot 0 & 0 \cdot (-5) + 7 \cdot 7 \end{bmatrix} = \begin{bmatrix} 4 & -45 \ 0 & 49 \end{bmatrix} \]

Step 3: Determine A^3 Existence

A^3 is defined if it can be computed as \( A \times A^2 \). Check matrix A: it is a 2x2 matrix. Since A^2 is calculated in Step 2, A^3 can be obtained by multiplying A by A^2.

Step 4: Calculate A^3

Multiply matrix A by matrix A^2 to compute A^3:\[A^3 = A \times A^2 = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} \times \begin{bmatrix} 4 & -45 \ 0 & 49 \end{bmatrix} = \begin{bmatrix} 2 \cdot 4 + (-5) \cdot 0 & 2 \cdot (-45) + (-5) \cdot 49 \ 0 \cdot 4 + 7 \cdot 0 & 0 \cdot (-45) + 7 \cdot 49 \end{bmatrix} = \begin{bmatrix} 8 & -385 \ 0 & 343 \end{bmatrix} \]

Key Concepts

Matrix Multiplication

Matrix multiplication is a way to combine two matrices to form a new one. This operation is only possible when the number of columns in the first matrix equals the number of rows in the second matrix. For instance, if you have a matrix

• with dimensions \( m \times n \) and another with dimensions \( n \times p \),

then their product will be a new matrix with dimensions \( m \times p \). Each element in the resulting matrix is calculated by taking the dot product of corresponding rows and columns from the two matrices.
In simpler terms, think of moving horizontally across the rows of the first matrix, and vertically down the columns of the second one, multiplying numbers and adding them up.
Remember this: matrix multiplication is not commutative, meaning \( A \times B \) is not the same as \( B \times A \).

Square Matrix

A square matrix is simply a matrix with the same number of rows and columns. For example, a 2x2 or a 3x3 matrix. The operations on square matrices are often simpler to handle because of this balance in dimensions.
Square matrices possess unique properties: they have a determinant, can be inverted (under certain conditions), and are often used in solving systems of linear equations.

• One important characteristic of square matrices is that they can be multiplied by themselves multiple times, leading to the calculation of exponents such as \( A^2 \), \( A^3 \), and so on.

In our exercise, matrix \(A\) is 2x2, making it a square matrix, which explains the feasibility of calculating its exponents, like \( A^2 \) and \( A^3 \).

Matrix Exponents

Matrix exponents involve multiplying a square matrix by itself a certain number of times. For example, \( A^2 \) means the matrix \( A \) is multiplied by itself once, while \( A^3 \) means it is multiplied by \( A^2 \).

• Keep in mind: matrix exponents are defined only for square matrices.
• Each resulting product has the potential to be calculated once you ensure the multiplication rules align.

Computing these exponents involves consecutively multiplying matrices, which follows similar rows and columns dot products as in normal matrix multiplication.
Matrix exponents are useful in various areas, such as in solving linear recurrence relations and modeling systems that evolve over time, known as dynamical systems.