Problem 26 - 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) \(D H\) (b) \(H D\)

Step-by-Step Solution

Answer

(a) \(D H = \begin{bmatrix} 27 & 4 \end{bmatrix}\); (b) \(H D\) cannot be performed.

Step 1: Determine Dimensions for DH

Matrix \(D\) is a \(1 \times 2\) matrix and matrix \(H\) is a \(2 \times 2\) matrix. When multiplying matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Here, \(D\) has 2 columns, and \(H\) has 2 rows, so we can multiply them. The resulting matrix will be \(1 \times 2\) from the rows of \(D\) and columns of \(H\).

Step 2: Compute DH

Multiply each element of the row of \(D\) with the corresponding elements of the columns in \(H\), and sum them to get each element of the new matrix. \[ D H = \begin{bmatrix} 7 & 3 \end{bmatrix} \begin{bmatrix} 3 & 1 \ 2 & -1 \end{bmatrix} \]= \[ \begin{bmatrix} (7 \times 3) + (3 \times 2) & (7 \times 1) + (3 \times -1) \end{bmatrix} \]= \[ \begin{bmatrix} 21 + 6 & 7 - 3 \end{bmatrix} \]= \[ \begin{bmatrix} 27 & 4 \end{bmatrix} \].

Step 3: Determine Dimensions for HD

Matrix \(H\) is a \(2 \times 2\) matrix and matrix \(D\) is a \(1 \times 2\) matrix. For matrix multiplication \(HD\) to be possible, \(H\) must have the same number of columns as \(D\) has rows. Here, \(H\) has 2 columns and \(D\) has 1 row, so the dimensions do not match, and the multiplication \(HD\) is not possible.

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial for performing matrix multiplication. A matrix is essentially an arrangement of numbers in rows and columns. The dimensions of a matrix are determined by the number of rows and the number of columns it has.
For instance, if you have a matrix with 2 rows and 3 columns, its dimensions are expressed as "2 x 3".
This concept is important because matrix multiplication depends on the dimensions of the matrices involved.

• When multiplying two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. If this condition is satisfied, the multiplication can be carried out.
• The resulting matrix's dimensions are determined by the number of rows from the first matrix and the number of columns from the second matrix.

Let's consider the example from the exercise: Matrix \(D\) has dimensions \(1 \times 2\) and matrix \(H\) has dimensions \(2 \times 2\). Here, the number of columns in \(D\) (2) matches the number of rows in \(H\) (2), making the multiplication \(DH\) feasible. As a result, the product \(DH\) will be of dimensions \(1 \times 2\). Understanding these fundamentals helps prevent errors during matrix operations.

Algebraic Operations

Algebraic operations with matrices mimic many of the operations we perform with numbers, although with some specific rules. A key operation is matrix multiplication, which allows us to combine matrices in a meaningful way.
In algebraic terms, matrix multiplication is like applying transformations to data, often used in areas such as systems equations or computer graphics. Let's delve into how this operation is performed.

• Element-wise multiplication: During matrix multiplication, each element of the rows from the first matrix is multiplied by the corresponding element of the columns in the second matrix. The products are then summed up to form an element of the resultant matrix.
• Order matters: Unlike regular multiplication, the order in which you multiply matrices is important. For instance, the multiplication order of \(DH\) is not the same as \(HD\), and it might not even be possible to compute both products, as shown in the exercise.

Applying these principles, the multiplication of matrix \(D\) by matrix \(H\) results in \[ \begin{bmatrix} 27 & 4 \end{bmatrix} \] by progressing through element-wise multiplication and addition.
Remember, ensuring the correct order and dimension compatibility is key to correctly performing algebraic operations with matrices.

Precaculus

Before venturing into calculus, a solid grasp of precalculus concepts, like matrices and basic algebra, is essential. Understanding these underlying principles prepares students for more advanced topics and their applications.

• Conceptual foundation: Concepts such as matrices organize data succinctly, providing a foundational tool for more complex mathematical systems encountered in calculus, like linear transformations and vector spaces.
• Operations practice: Mastering operations such as matrix addition, subtraction, and importantly, multiplication, lays the groundwork for calculus-based applications. These operations are fundamental skills needed in higher-level mathematics.
• Problem-solving: Matrices train students to think analytically and solve problems step-by-step, a critical skill for calculus that evolves into tackling integrals and derivatives.

Engaging with matrices and their operations, such as those in the provided exercise, not only solidifies your algebraic skills but also dramatically enhances your readiness to tackle calculus challenges.
So, having a strong foundation in these concepts aids the transition from basic operations to the more robust applications found in calculus.