Problem 27 - 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) \(A D\) (b) \(D A\)

Step-by-Step Solution

Answer

AD is not defined; DA = \(\begin{bmatrix} 14 & -14 \end{bmatrix}\).

Step 1: Check Matrices Multiplication Compatibility for AD

To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. Matrix \(A\) has dimensions \(2 \times 2\), and matrix \(D\) has dimensions \(1 \times 2\). The number of columns in \(A\) is 2, and the number of rows in \(D\) is 1. Since these do not match, matrix multiplication \(AD\) is not defined.

Step 2: Check Matrices Multiplication Compatibility for DA

Again, the compatibility of matrix multiplication requires the number of columns in the first matrix to match the number of rows in the second one. Matrix \(D\) has dimensions \(1 \times 2\), and \(A\) has dimensions \(2 \times 2\). Here, the number of columns in \(D\) is 2, which matches with the number of rows in \(A\), allowing the multiplication \(DA\).

Step 3: Perform Matrix Multiplication for DA

With \(D = \begin{bmatrix} 7 & 3 \end{bmatrix}\) and \(A = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix}\), we perform the multiplication. \[ DA = \begin{bmatrix} 7 & 3 \end{bmatrix} \cdot \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} = \begin{bmatrix} (7 \cdot 2 + 3 \cdot 0) & (7 \cdot -5 + 3 \cdot 7) \end{bmatrix} = \begin{bmatrix} 14 & -14 \end{bmatrix} \]

Step 4: Interpret the Results

The multiplication \(AD\) cannot be performed due to a mismatch in the matrix dimensions. However, the multiplication \(DA\) is valid and results in the matrix \(\begin{bmatrix} 14 & -14 \end{bmatrix}\).

Key Concepts

Matrix Dimensions

Understanding the dimensions of a matrix is crucial when dealing with matrix operations. A matrix is typically described in terms of its dimensions, which are given in the form \(m \times n\), where \(m\) represents the number of rows and \(n\) represents the number of columns.
For example, matrix \(A\) is a \(2 \times 2\) matrix because it has 2 rows and 2 columns. Meanwhile, matrix \(D\) is \(1 \times 2\) because it consists of 1 row and 2 columns.

When identifying matrices by their dimensions, you can quickly infer the matrix's shape and size. This identification helps set the stage for determining the compatibility of matrices for operations such as multiplication. Being familiar with the dimensions provides a clearer understanding of how matrices can interact through various operations.

Matrix Compatibility

Matrix compatibility, especially in the context of multiplication, relies heavily on the dimensions of the matrices involved.

If you wish to multiply two matrices, say matrix \(A\) and matrix \(B\), a crucial compatibility condition must be met: the number of columns in matrix \(A\) must equal the number of rows in matrix \(B\). This compatibility condition allows for the rows of \(A\) to interact with the columns of \(B\) during multiplication.

• For the matrices in our exercise: matrix \(A\) is \(2 \times 2\), while matrix \(D\) is \(1 \times 2\). This means that \(AD\) is not compatible because 2 columns from matrix \(A\) do not match 1 row from matrix \(D\).
• Conversely, \(D\) and \(A\) can be multiplied in the order \(DA\) because \(D\) has 2 columns, matching \(A\)'s 2 rows.

Matrices that do not meet this condition cannot be multiplied, highlighting the importance of understanding matrix compatibility.

Matrix Operations

Matrix operations, such as multiplication, involve a specific set of rules and procedures to carry out calculations efficiently. When performing matrix multiplication, keep in mind how each element is computed.

To multiply compatible matrices, take each row from the first matrix and multiply it by each column of the second matrix, summing the products for each position in the resulting matrix.
For the example given in the exercise with \(DA\), we have:

• Matrix \(D = \begin{bmatrix} 7 & 3 \end{bmatrix}\)
• Matrix \(A = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix}\)

This results in matrix multiplication where:

• For the first position, calculate \((7 \cdot 2 + 3 \cdot 0) = 14\).
• For the second position, calculate \((7 \cdot -5 + 3 \cdot 7) = -14\).

Thus, the product \(DA\) is \(\begin{bmatrix} 14 & -14 \end{bmatrix}\). Understanding these operational mechanics is essential for carrying out correct and meaningful matrix operations.