Problem 28 - 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) \(D H\) (b) \(H D\)

Step-by-Step Solution

Answer

(a) \(D H = \left[\begin{array}{rr}27 & 4\end{array}\right]\). (b) \(H D\) cannot be performed.

Step 1: Identify the Dimensions of D and H

Matrix \(D\) has dimensions \(1 \times 2\), as it has 1 row and 2 columns. Matrix \(H\) has dimensions \(2 \times 2\), as it has 2 rows and 2 columns.

Step 2: Determine the Possibility of Multiplying D and H

For matrix multiplication \(D H\), the number of columns in \(D\) must match the number of rows in \(H\). Matrix \(D\) has 2 columns and matrix \(H\) has 2 rows, so \(D H\) is a valid operation. The result will be a \(1 \times 2\) matrix.

Step 3: Calculate the Matrix Product D H

Perform the multiplication: \[ D \times H = \left[\begin{array}{lll}7 & 3\end{array}\right] \times \left[\begin{array}{rr}3 & 1 \ 2 & -1\end{array}\right] \]Compute each element of the resulting matrix. First element: \(7 \times 3 + 3 \times 2 = 21 + 6 = 27\).Second element: \(7 \times 1 + 3 \times (-1) = 7 - 3 = 4\).So, \(D H = \left[\begin{array}{rr}27 & 4\end{array}\right]\).

Step 4: Determine the Possibility of Multiplying H and D

For matrix multiplication \(H D\), the number of columns in \(H\) must match the number of rows in \(D\). Matrix \(H\) has 2 columns and matrix \(D\) has 1 row, so \(H D\) is not a valid operation.

Step 5: Conclusion about H D

Since \(H D\) is not a valid operation due to mismatched dimensions, it cannot be performed.

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial for performing matrix operations. A matrix's dimensions are expressed as "rows × columns". For example, if a matrix has 2 rows and 3 columns, it is referred to as a 2 × 3 matrix. This information is vital when determining whether two matrices can be multiplied or not. Matrix multiplication is only possible when the number of columns in the first matrix matches the number of rows in the second matrix.

Let's consider the matrices mentioned in the exercise:

• Matrix \(D\) is a \(1 \times 2\) matrix, having 1 row and 2 columns.
• Matrix \(H\) is a \(2 \times 2\) matrix, with 2 rows and 2 columns.

The dimensions tell us what matrix operations are valid, guiding us to see that \(DH\) can be computed, whereas \(HD\) cannot because their dimensions do not align correctly for multiplication.

Matrix Algebra

Matrix algebra involves various operations that rely on the system of matrix dimensions and element-wise calculations. Apart from basic addition and subtraction, matrices involve more complex operations like multiplication, which involves particular rules based on matrix dimensions.

In matrix algebra, when calculating the product of matrices, it's essential to consider the alignment of their dimensions. Once we verify that they can be multiplied, we compute the product by calculating the sum of the products of their corresponding elements row by column. For instance, in the computation of \(DH\), each element in the resulting matrix is obtained by summing the products of the elements in the rows of \(D\) and columns of \(H\). This precise arithmetic simplifies to a structured process evident in matrix multiplication.

Matrix Operations

Matrix operations such as multiplication have their unique set of rules compared to normal arithmetic. One core rule is the dimension alignment which dictates the feasibility of the operation. Already mentioned, the number of columns in the first matrix must match the number of rows in the second matrix. This condition was satisfied in the operation \(DH\) given in the exercise, resulting in a new matrix of dimension \(1 \times 2\).

For \(DH\):

• Calculate the first element by multiplying elements of row 1 of \(D\) with column 1 of \(H\): \(7 \times 3 + 3 \times 2 = 27\).
• Calculate the second element by multiplying elements of row 1 of \(D\) with column 2 of \(H\): \(7 \times 1 + 3 \times (-1) = 4\).

However, \(HD\) is not possible due to dimension mismatch, as matrix \(H\) has 2 columns, and matrix \(D\) needs 2 rows, not 1, to be a valid operation. Such mathematical checks are fundamental in matrix operations, ensuring the resulting calculations are correct and meaningful.