Problem 29 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

The matrix \((DA)B\) equals d\[\begin{bmatrix} 28 & 21 & 28 \end{bmatrix}\].

Step 1: Matrix Dimensions Check for DA

Matrix \(D\) is \(1 \times 2\) and matrix \(A\) is \(2 \times 2\). To multiply these matrices, the inner dimensions must match, i.e., the number of columns in \(D\) must equal the number of rows in \(A\). Since they both are 2, the multiplication \(DA\) is possible, producing a \(1 \times 2\) matrix.

Step 2: Calculate DA

To compute \(DA\), we perform matrix multiplication:\[D = \begin{bmatrix} 7 & 3 \end{bmatrix},\quad A = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix}\]Perform the multiplication:\[DA = \begin{bmatrix} 7 & 3 \end{bmatrix} \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} = \begin{bmatrix} (7 \times 2 + 3 \times 0) & (7 \times -5 + 3 \times 7) \end{bmatrix} = \begin{bmatrix} 14 & -14 \end{bmatrix}\]

Step 3: Matrix Dimensions Check for (DA)B

Now that \(DA\) is a \(1 \times 2\) matrix, we need to check if we can multiply it with \(B\), which is a \(2 \times 3\) matrix. The number of columns in \(DA\) (which is 2) must match the number of rows in \(B\) (which is also 2). Hence, the multiplication \((DA)B\) is possible, resulting in a \(1 \times 3\) matrix.

Step 4: Calculate (DA)B

Now calculate \((DA)B\) using matrix multiplication:\[DA = \begin{bmatrix} 14 & -14 \end{bmatrix},\quad B = \begin{bmatrix} 3 & \frac{1}{2} & 5 \ 1 & -1 & 3 \end{bmatrix}\]Perform the multiplication:\[(DA)B = \begin{bmatrix} 14 & -14 \end{bmatrix} \begin{bmatrix} 3 & \frac{1}{2} & 5 \ 1 & -1 & 3 \end{bmatrix} \]This yields:\[= \begin{bmatrix} (14 \times 3 + (-14) \times 1) & (14 \times \frac{1}{2} + (-14) \times -1) & (14 \times 5 + (-14) \times 3) \end{bmatrix} \]\[= \begin{bmatrix} 28 & 21 & 28 \end{bmatrix} \]

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is essential when working with matrices, especially for operations like multiplication. The dimensions of a matrix are described by the number of rows and columns it has. For instance, if a matrix has two rows and three columns, it is a \(2 \times 3\) matrix. Matrices are often displayed as a rectangular array of numbers, where each entry belongs to a specific row and column.

• For matrix multiplication to be defined, the inner dimensions must match. This means that if you have a matrix \(A\) with dimensions \(m \times n\), it can only be multiplied by a matrix \(B\) if \(B\) has dimensions \(n \times p\). The resulting matrix will have dimensions \(m \times p\).
• Considering the given example, matrix \(D\) is a \(1 \times 2\) matrix, and matrix \(A\) is a \(2 \times 2\) matrix. Since the number of columns in \(D\) is equal to the number of rows in \(A\), the multiplication \(DA\) is possible, resulting in a \(1 \times 2\) matrix.

Having an understanding of matrix dimensions ensures that you know whether particular operations can be performed, avoiding futile calculations.

Algebraic Operations

Algebraic operations with matrices involve mathematical computations such as addition, subtraction, multiplication, and sometimes division, though matrix division is complex and typically involves inverses.

• Matrix multiplication, a key algebraic operation, is not as straightforward as addition or subtraction. It involves multiplying rows by columns across two matrices.
• To multiply matrices, follow these steps: Take each element of a row from the first matrix and multiply it by the corresponding element of a column from the second matrix, then sum these products to get an entry in the resulting matrix.

In our step 2 example, matrix \(D\) is multiplied by matrix \(A\). The resultant matrix \(DA\) is calculated by summing the products for each row-column interaction, giving a final output matrix of \(\begin{bmatrix} 14 & -14 \end{bmatrix}\). This approach is repeated for each element when multiplying \((DA)B\) to get the final result \(\begin{bmatrix} 28 & 21 & 28 \end{bmatrix}\).

Matrix Algebra

Matrix algebra is a foundational topic in linear algebra with wide applications in various fields, including computer graphics, statistics, and engineering. The primary foundation of matrix algebra is handling vectors and transforming them efficiently.

• Matrices simplify complex systems of linear equations by providing a concise notation to manipulate and solve them.
• Operations like matrix multiplication, addition, and finding inverses play a crucial role in broader mathematical computations.
• Matrix algebra also often involves computing determinants and eigenvalues, which provide insights into the properties of matrices.

By learning matrix algebra, a student builds the skills to perform intermediate and advanced mathematical tasks with matrices. There is a systematic method to these operations, like ensuring correct dimensions for multiplication and using precise calculations, as demonstrated in both parts of the example \((DA)B\). This consistency in methodical processes underscores why matrix algebra is a powerful mathematical tool for problem-solving.