Problem 32 - 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) \((D A) B\) (b) \(D(A B)\)

Step-by-Step Solution

Answer

(DA)B = [65, 0.5, 105]; D(AB) = [28, 28, 126].

Step 1: Check Matrix Sizes for (DA)B

Matrix \(D\) is \(1 \times 2\), and matrix \(A\) is \(2 \times 2\). We can multiply \(D\) by \(A\) because the number of columns in \(D\) is equal to the number of rows in \(A\), resulting in a \(1 \times 2\) matrix. Next, check if this product can be multiplied by \(B\). Matrix \(B\) is \(2 \times 3\), so \((DA)B\) is possible, resulting in a \(1 \times 3\) matrix.

Step 2: Calculate DA

Calculate \(DA\) by multiplying matrices \(D\) and \(A\):\[D \times A = \left[ \begin{array}{ll}7 & 3\end{array} \right] \times \left[ \begin{array}{rr}2 & -5 \0 & 7\end{array} \right] = \left[ \begin{array}{ll}14 & r21\end{array} \right]\] where, \(14 = 7 \times 2 + 3 \times 0\) and \(21 = 7 \times (-5) + 3 \times 7\).

Step 3: Calculate (DA)B

Multiply the result \( \text{DA} = \left[ 14 \quad 21 \right] \) by \(B\):\[\left[ \begin{array}{ll} 14 & 21 \end{array} \right] \times \left[ \begin{array}{rrr} 3 & \frac{1}{2} & 5 \ 1 & -1 & 3 \end{array} \right] = \left[ \begin{array}{lll} 65 & \frac{1}{2} & 105 \end{array} \right]\]Here, each element is calculated as follows: \(65 = 14 \times 3 + 21 \times 1\), \(0.5 = 14 \times 0.5 - 21 \times 1\), and \(105 = 14 \times 5 + 21 \times 3\).

Step 4: Check Matrix Sizes for D(AB)

First, calculate \(AB\). Matrix \(A\) is \(2 \times 2\), and matrix \(B\) is \(2 \times 3\). The result will be a \(2 \times 3\) matrix. Next, check if matrix \(D\) (\(1 \times 2\)) can multiply the result \(AB\). Since \(D\) needs a \(2 \times \_)\) matrix to multiply with, we can proceed.

Step 5: Calculate AB

Multiply matrices \(A\) and \(B\):\[A \times B = \left[ \begin{array}{rr} 2 & -5 \ 0 & 7 \end{array} \right] \times \left[ \begin{array}{rrr} 3 & \frac{1}{2} & 5 \ 1 & -1 & 3 \end{array} \right] = \left[ \begin{array}{rrr} 1 & 7 & 15 \ 7 & -7 & 21 \end{array} \right]\]

Step 6: Calculate D(AB)

Multiply matrix \(D\) by the result \(AB\):\[D \times (AB) = \left[ \begin{array}{ll} 7 & 3 \end{array} \right] \times \left[ \begin{array}{rrr} 1 & 7 & 15 \ 7 & -7 & 21 \end{array} \right] = \left[ \begin{array}{rrr} 28 & 28 & 126 \end{array} \right]\]This results from calculating: \(28 = 7 \times 1 + 3 \times 7\), \(28 = 7 \times 7 + 3 \times (-7)\), \(126 = 7 \times 15 + 3 \times 21\).

Key Concepts

Matrix Dimensions

Understanding the dimensions of matrices is crucial in matrix multiplication. A matrix's dimensions are represented as rows x columns, which helps in determining whether two matrices can be multiplied. For example, a matrix consisting of 2 rows and 3 columns is a 2x3 matrix.

When you multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. For example, if you have a 2x3 matrix and a 3x2 matrix, the multiplication is possible, and the resulting matrix will have dimensions of 2x2.

Let’s take matrices D and A from the exercise: D is a 1x2 matrix, and A is a 2x2 matrix. They can be multiplied since the number of columns in D (2) equals the number of rows in A (2). The product is a matrix with dimensions of 1x2. This foundation of matrix dimensions is critical for performing correct matrix multiplications.

Matrix Product

The matrix product, or matrix multiplication, combines two matrices to form a new matrix. To compute this, each element in the resulting matrix is obtained by taking the dot product of rows from the first matrix with columns from the second. This might sound complex at first, but it involves straightforward arithmetic.

Consider the multiplication of matrices D and A from our example. Matrix D is 1x2 and matrix A is 2x2. For the first column of the resultant matrix, multiply each entry of the 1st row of D with corresponding entries of each column of A and sum these products:

• First element: (7x2) + (3x0) = 14
• Second element: (7x(-5)) + (3x7) = 21

The concept of matrix product is a building block for more complex algebraic operations involving matrices, making it vital to master.

Algebraic Operations

Matrix algebraic operations extend the basic arithmetic operations to matrices. These include addition, subtraction, and multiplication. However, unlike numbers, not every matrix operation always makes sense mathematically.

In the context of our problem, we are interested in the acceptable sequence of operations. For matrix addition or subtraction, both matrices must have identical dimensions, which is not a concern in our exercise. Here, we focus on multiplication, notably illustrated by $(DA)B$ and $D(AB)$.

When dealing with different sequences of multiplication, note that matrix multiplication is associative. This means that no matter how matrices are grouped in multiplication (as in $(DA)B$ versus $D(AB)$), the result remains the same, provided the multiplication is defined for all intermediate operations.

In both scenarios of our exercise, the algebraic operations rely on the matrix dimensions allowing the possible multiplication drills at each step. Ultimately, understanding and performing these operations accurately enables you to handle larger systems and problems in advanced mathematics and applied sciences.