Problem 29 - Precalculus: Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

(a) \(A H = \begin{bmatrix} -4 & 7 \\ 14 & -7 \end{bmatrix}\) (b) \(H A = \begin{bmatrix} 6 & -8 \\ 4 & -17 \end{bmatrix}\)

Step 1: Determine if AH is Possible

To multiply matrices \(A\) and \(H\), the number of columns in \(A\) must equal the number of rows in \(H\). \(A\) is a \(2\times 2\) matrix, and \(H\) is a \(2\times 2\) matrix. Both have 2 columns and 2 rows respectively, hence the multiplication is possible.

Step 2: Compute Product AH

Multiply each row of matrix \(A\) by each column of matrix \(H\):\[A H = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} \begin{bmatrix} 3 & 1 \ 2 & -1 \end{bmatrix}\]- For the first row, first column: \(2 \times 3 + (-5) \times 2 = 6 - 10 = -4\)- For the first row, second column: \(2 \times 1 + (-5) \times (-1) = 2 + 5 = 7\)- For the second row, first column: \(0 \times 3 + 7 \times 2 = 0 + 14 = 14\)- For the second row, second column: \(0 \times 1 + 7 \times (-1) = 0 - 7 = -7\)So, \(A H = \begin{bmatrix} -4 & 7 \ 14 & -7 \end{bmatrix}\).

Step 3: Determine if HA is Possible

To multiply matrices \(H\) and \(A\), the number of columns in \(H\) must equal the number of rows in \(A\). Both matrices are \(2\times 2\), so the multiplication is possible.

Step 4: Compute Product HA

Multiply each row of matrix \(H\) by each column of matrix \(A\):\[H A = \begin{bmatrix} 3 & 1 \ 2 & -1 \end{bmatrix} \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix}\]- For the first row, first column: \(3 \times 2 + 1 \times 0 = 6 + 0 = 6\)- For the first row, second column: \(3 \times (-5) + 1 \times 7 = -15 + 7 = -8\)- For the second row, first column: \(2 \times 2 + (-1) \times 0 = 4 + 0 = 4\)- For the second row, second column: \(2 \times (-5) + (-1) \times 7 = -10 - 7 = -17\)So, \(H A = \begin{bmatrix} 6 & -8 \ 4 & -17 \end{bmatrix}\).

Key Concepts

Matrix Algebra

Matrix algebra involves expressions and operations with matrices, which are rectangular arrays of numbers or variables arranged in rows and columns. In the context of the exercise provided, matrix algebra is primarily concerned with adding, subtracting, and multiplying matrices.

These operations follow specific rules that depend on the dimensions of the matrices involved. For multiplication, a key point in matrix algebra is checking that the number of columns in the first matrix is equal to the number of rows in the second matrix. This ensures that the matrices are conformable for multiplication.

Matrix algebra is versatile and widely used in various fields such as physics, computer science, and engineering. It's crucial to keep in mind the dimensions of the matrices, always ensuring they align for the operations you want to perform.

Matrices

Matrices are foundational to matrix algebra. A matrix is essentially a grid that organizes numbers in a specific way, defined by rows and columns. For instance, matrix \(A\) from the exercise is a \(2 \times 2\) matrix, meaning it has 2 rows and 2 columns, and looks like this:

• A matrix (\( A \)) with elements arranged as \[\begin{bmatrix}2 & -5 \0 & 7\end{bmatrix}\]

Each number, or element, within a matrix is identified by its position given by row and column indices.

Matrices can represent systems of linear equations, transformations, or other abstract data concepts. Understanding how to interpret and manipulate matrices through operations like addition (element-wise) and multiplication (as per specific algebraic rules) is key to mastering their application.

In practical applications, matrices are used in everything from representing data in a computer to simulating physical environments in video games.

Matrix Operations

Matrix operations are specific calculations performed on matrices, like the exercise challenge you to do with matrices \( A \) and \( H \). The primary operations are addition, subtraction, and multiplication. In the realm of matrices:

• **Addition and Subtraction**: Only matrices of the same dimension can be added or subtracted by performing operations on corresponding elements.
• **Multiplication**: This involves taking each element of a row from the first matrix and multiplying it with each element of a column in the second matrix, then summing up these products. This was done in both \( AH \) and \( HA \) calculations from the exercise. The multiplication result is a matrix whose number of rows equals the number of rows of the first matrix and the number of columns of the second matrix.

For multiplication, remember:

• The element in the \((i, j)\) position (i.e., in the ith row and jth column) of the product matrix is the dot product of the ith row of the first matrix and the jth column of the second matrix.

Note: Matrix multiplication is not commutative, meaning \( AH \) might differ from \( HA \) in both elements and dimensions if applicable.
Mastering these operations is essential as they form the backbone of matrix algebra, used in everything from solving mathematical models to processing image data in cameras.