Problem 35 - 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) \(ABE\) (b) \(AHE\)

Step-by-Step Solution

Answer

(a) Result: \( \begin{bmatrix} 18 \\ -7 \end{bmatrix} \), (b) Not defined.

Step 1: Determine Dimensions of Matrices for Part (a)

The dimensions of the matrices are as follows: \(A\) is a \(2 \times 2\) matrix, \(B\) is a \(2 \times 3\) matrix, and \(E\) is a \(3 \times 1\) matrix. We first check if these matrix multiplications are defined: \(AB\) is possible since the number of columns in \(A\) matches the number of rows in \(B\) (2), yielding a \(2 \times 3\) matrix. For \((AB)E\) to be defined, the number of columns in \(AB\) must match the rows in \(E\) (3), which they do. Thus, \(ABE\) is defined.

Step 2: Multiply Matrices A and B

Compute \(AB\) where \(A = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix}\) and \(B = \begin{bmatrix} 3 & \frac{1}{2} & 5 \ 1 & -1 & 3 \end{bmatrix}\). Perform the multiplication: \[ AB = \begin{bmatrix} (2 \cdot 3 + (-5) \cdot 1) & (2 \cdot \frac{1}{2} + (-5) \cdot (-1)) & (2 \cdot 5 + (-5) \cdot 3) \ (0 \cdot 3 + 7 \cdot 1) & (0 \cdot \frac{1}{2} + 7 \cdot (-1)) & (0 \cdot 5 + 7 \cdot 3) \end{bmatrix} \]\[ = \begin{bmatrix} 1 & \frac{17}{2} & -5 \ 7 & -7 & 21 \end{bmatrix} \].

Step 3: Multiply Result of AB with E

Multiply \(AB\) by \(E\), where \(E = \begin{bmatrix} 1 \ 2 \ 0 \end{bmatrix}\):\[ (AB)E = \begin{bmatrix} 1 & \frac{17}{2} & -5 \ 7 & -7 & 21 \end{bmatrix} \begin{bmatrix} 1 \ 2 \ 0 \end{bmatrix} \] Compute the multiplication by summing products for corresponding elements:\[(AB)E = \begin{bmatrix} 1 \cdot 1 + \frac{17}{2} \cdot 2 + (-5) \cdot 0 \ 7 \cdot 1 + (-7) \cdot 2 + 21 \cdot 0 \end{bmatrix} = \begin{bmatrix} 18 \ -7 \end{bmatrix} \].

Step 4: Determine Dimensions of Matrices for Part (b)

Now consider \(AHE\). \(A\) is a \(2 \times 2\) matrix, \(H\) is a \(2 \times 2\) matrix, and \(E\) is a \(3 \times 1\) matrix. Firstly, we must compute \(AH\), which is possible because both have compatible dimensions (\(2 \times 2\)). This results in a \(2 \times 2\) matrix. However, this matrix cannot be multiplied by \(E\) (\(3 \times 1\)) because their dimensions do not align, so the operation \(AHE\) is not defined.

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial in determining whether matrix operations, like multiplication, can be carried out. A matrix's dimensions are defined by its number of rows (horizontal lines) and columns (vertical lines). For example, if a matrix has three rows and four columns, its dimensions are expressed as \(3 \times 4\). When multiplying two matrices, an important rule is that the number of columns in the first matrix must be equal to the number of rows in the second. Let's look at an example:

• Matrix \(A\) is \(2 \times 2\), meaning it has 2 rows and 2 columns.
• Matrix \(B\) is \(2 \times 3\).

To multiply \(A\) by \(B\), the number of columns in \(A\) (which is 2) must match the number of rows in \(B\) (also 2). If this condition is met, the resulting product, \(AB\), will have the number of rows of \(A\) and the number of columns of \(B\), making it a \(2 \times 3\) matrix. This understanding helps explain why some matrix multiplications can be performed while others cannot, such as the exercise where \((AB)E\) was defined, but \(AHE\) was not.

Matrix Operations

Matrix operations involve various calculations that you can perform on matrices, including addition, subtraction, and multiplication. The focus here is on multiplication, which requires a deep understanding due to its rules and effects.Matrix multiplication involves taking rows from the first matrix and columns from the second matrix to calculate an element in the resulting product matrix. For example, to find the element in the first row, first column of the resulting matrix when multiplying matrices \(A\) and \(B\), you take the first row of \(A\) and the first column of \(B\):

• You multiply each element of the row by the corresponding element of the column.
• Then, you sum these products.

This process is repeated for each element of the resulting matrix.The example in the exercise illustrates this operation with matrices \(A\) and \(B\), leading to the product \(AB\) that was then multiplied by matrix \(E\). Each step requires care, as any mistake in multiplication or addition could lead to incorrect results. It emphasizes the importance of accuracy in such operations.

Matrix Algebra

Matrix algebra is a powerful mathematical tool, enabling the manipulation of datasets in a structured format. It spans various operations, with matrix multiplication being a key one due to its application in numerous fields like computer graphics, scientific computing, and data analysis.In matrix algebra, operations follow specific laws and properties:

• Associative Property: The product of matrices is associative, which means \((AB)C = A(BC)\).
• Distributive Property: Matrix multiplication is distributive over addition, such as \(A(B+C) = AB + AC\).
• Non-Commutative: Unlike numbers, matrix multiplication is generally not commutative, meaning \(AB\) is not necessarily equal to \(BA\).

The exercise demonstration emphasizes the associative property since the problem involved computing \((AB)E\). Understanding these properties not only aids in performing correct operations but also in recognizing when certain computations may be undefined, such as when needing to multiply matrices with incompatible dimensions, like \(AHE\). Mastering these concepts allows for efficient and effective algebraic manipulation of matrix data.