Problem 30 - 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 result of the matrix operation \(D(AB)\) is \(\begin{bmatrix} 28 & 21 & 28 \end{bmatrix}\).

Step 1: Calculate AB

The first step is to calculate the product of matrices \(A\) and \(B\). Matrix \(A\) is a \(2 \times 2\) matrix, and \(B\) is a \(2 \times 3\) matrix. Since the number of columns in \(A\) is equal to the number of rows in \(B\), we can multiply them. The resulting matrix \(AB\) will be a \(2 \times 3\) matrix. Perform the multiplication as follows:\[AB = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} \times \begin{bmatrix} 3 & \frac{1}{2} & 5 \ 1 & -1 & 3 \end{bmatrix} = \begin{bmatrix} (2)(3) + (-5)(1) & (2)(\frac{1}{2}) + (-5)(-1) & (2)(5) + (-5)(3) \ (0)(3) + (7)(1) & (0)(\frac{1}{2}) + (7)(-1) & (0)(5) + (7)(3) \end{bmatrix} \]Calculate each entry to get:\[AB = \begin{bmatrix} 6 - 5 & 1 + 5 & 10 - 15 \ 0 + 7 & 0 - 7 & 0 + 21 \end{bmatrix} = \begin{bmatrix} 1 & 6 & -5 \ 7 & -7 & 21 \end{bmatrix}\]

Step 2: Check Compatibility for D(AB)

Next, check if the multiplication \(D(AB)\) is possible. Matrix \(D\) is a \(1 \times 2\) matrix, and the resulting matrix \(AB\) is a \(2 \times 3\) matrix from Step 1. The multiplication is possible since the number of columns in \(D\) (which is 2) is the same as the number of rows in \(AB\) (which is 2). The resulting matrix will be a \(1 \times 3\) matrix.

Step 3: Calculate D(AB)

Multiply matrix \(D\) and matrix \(AB\) using the following calculation:\[D(AB) = \begin{bmatrix} 7 & 3 \end{bmatrix} \times \begin{bmatrix} 1 & 6 & -5 \ 7 & -7 & 21 \end{bmatrix} = \begin{bmatrix} (7)(1) + (3)(7) & (7)(6) + (3)(-7) & (7)(-5) + (3)(21) \end{bmatrix}\]Calculate each entry to get:\[D(AB) = \begin{bmatrix} 7 + 21 & 42 - 21 & -35 + 63 \end{bmatrix} = \begin{bmatrix} 28 & 21 & 28 \end{bmatrix}\]

Step 4: Conclusion

The final result of multiplying \(D\) by \( (AB) \) is a \(1 \times 3\) matrix:\[D(AB) = \begin{bmatrix} 28 & 21 & 28 \end{bmatrix}\]

Key Concepts

Matrix Algebra

Matrix algebra is the way to handle matrices that are more than just numbers; they are arrays of numbers that can represent complex data or systems of linear equations. When you think of matrix algebra, think of performing arithmetic on matrices, just like you do with simple numbers. However, matrix algebra follows specific rules. Not every operation you can do with ordinary numbers applies directly to matrices.
For instance, when you multiply two matrices,

• the order of multiplication matters
• you need to ensure that the number of columns in the first matrix matches the number of rows in the second matrix
• the resulting matrix takes its dimensions from the outer dimensions of the multiplied matrices

Matrix addition and subtraction are a bit simpler, as they just require that the matrices are of the same size. Learn these foundational rules, and matrix algebra becomes a powerful tool for many areas in math, engineering, and science.

Matrix Dimensions

Understanding matrix dimensions is crucial in matrix operations and algebra. A matrix is essentially a table of numbers arranged in rows and columns, and the dimensions of a matrix are given as 'rows × columns'. For example, matrix \(A\) is a \(2 \times 2\) matrix, meaning it has 2 rows and 2 columns.
The dimensions are important because they determine if two matrices can be multiplied. If you have matrix \(A\) of size \(m \times n\) and matrix \(B\) of size \(p \times q\), they can be multiplied only if \(n = p\). The resulting matrix will then have dimensions \(m \times q\).

• If the dimensions do not align for multiplication, then the operation cannot be performed.
• The dimensions of the result are always determined by the number of rows of the first matrix and the number of columns of the second matrix.

Once you master understanding matrix dimensions, you'll find it much easier to manipulate and solve problems involving matrices.

Matrix Operations

Matrix operations include various computations you can achieve using matrices, such as addition, subtraction, and multiplication. They are similar to the operations you do with numbers, but matrices follow specific conditions and rules.
For multiplication, each entry in the resulting matrix is calculated as the sum of products of corresponding entries from a row of the first matrix and a column of the second matrix. It is especially important to

• follow the rule that the number of columns in the first matrix must be equal to the number of rows in the second matrix
• apply the multiplication formula correctly

For example, in the exercise, multiplying matrix \(D\) by the product \(AB\) requires each element in \(D(AB)\) to be derived from a sum of products. Beyond multiplication, remember:

• Addition and subtraction require matrices of the same dimensions, where you simply add or subtract each corresponding element.
• Matrix transposition involves flipping a matrix over its diagonal, swapping the row and column indices of each element.

By understanding these operations, you unlock the ability to model and solve complex equations and manipulate data efficiently in matrix form.