Problem 29 - Algebra and Trigonometry

Question

The matrices \(A, B, C, D, E, F,\) and \(G\) are defined as follows. $$A=\left[\begin{array}{rr}{2} & {-5} \\ {0} & {7}\end{array}\right] \quad B=\left[\begin{array}{rrr}{3} & {\frac{1}{2}} & {5} \\ {1} & {-1} & {3}\end{array}\right] \quad C=\left[\begin{array}{rrr}{2} & {-\frac{5}{2}} & {0} \\ {0} & {2} & {-3}\end{array}\right]$$ $$\begin{array}{l}{D=\left[\begin{array}{lll}{7} & {3}\end{array}\right]} & {E=\left[\begin{array}{lll}{1} \\ {1} \\ {2} \\ {0}\end{array}\right]} \\\ {F=\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right] \quad G=\left[\begin{array}{rrr}{5} & {-3} & {10} \\\ {6} & {1} & {0} \\ {-5} & {2} & {2}\end{array}\right]}\end{array}$$ Carry out the indicated algebraic operation, or explain why it cannot be performed. $$ (D A) B $$

Step-by-Step Solution

Answer

The calculation \((DA)B\) results in the matrix \([28, 21, 28]\).

Step 1: Identify Dimensions

First, let's identify the dimensions of the matrices involved in the operation \((D A) B\). The matrix \(D\) has a dimension of \(1 \times 2\). The matrix \(A\) is \(2 \times 2\). The matrix \(B\) is \(2 \times 3\).

Step 2: Determine Compatibility for Multiplication DA

Matrix multiplication is possible when the number of columns in the first matrix matches the number of rows in the second matrix. Here, matrix \(D\) (\(1 \times 2\)) can multiply with matrix \(A\) (\(2 \times 2\)) since the number of columns in \(D\) (2) matches the number of rows in \(A\) (2).

Step 3: Perform Multiplication DA

Now, compute the product \(DA\). Multiplying \(D = \begin{bmatrix} 7 & 3 \end{bmatrix}\) by \(A = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix}\) yields:\[DA = \begin{bmatrix} 7 \cdot 2 + 3 \cdot 0 & 7 \cdot (-5) + 3 \cdot 7 \end{bmatrix} = \begin{bmatrix} 14 & -35 + 21 \end{bmatrix} = \begin{bmatrix} 14 & -14 \end{bmatrix}\]

Step 4: Determine Compatibility for Multiplication (DA)B

The resulting matrix \(DA\) has a dimension of \(1 \times 2\). The matrix \(B\) has a dimension of \(2 \times 3\). Since the number of columns in \(DA\) (2) matches the number of rows in \(B\) (2), they can be multiplied together.

Step 5: Perform Multiplication (DA)B

Compute the product \((DA)B\). Multiply \(DA = \begin{bmatrix} 14 & -14 \end{bmatrix}\) with \(B = \begin{bmatrix} 3 & \frac{1}{2} & 5 \ 1 & -1 & 3 \end{bmatrix}\):\[(DA)B = \begin{bmatrix} 14 \cdot 3 + (-14) \cdot 1 & 14 \cdot \frac{1}{2} + (-14) \cdot (-1) & 14 \cdot 5 + (-14) \cdot 3 \end{bmatrix}\]\[= \begin{bmatrix} 42 - 14 & 7 + 14 & 70 - 42 \end{bmatrix} = \begin{bmatrix} 28 & 21 & 28 \end{bmatrix}\]

Key Concepts

Matrix Dimensions

Matrix dimensions are a fundamental concept in matrix algebra that helps to describe the size and shape of a matrix. A matrix is essentially a rectangular array of numbers arranged in rows and columns. To specify the dimensions of a matrix, you mention the number of rows first and then the number of columns. For example, if a matrix has 2 rows and 3 columns, it is said to be a "2 × 3 matrix".

Understanding the dimensions allows you to accurately represent and manipulate the data the matrix contains. For instance, in the original exercise, matrix \(A\) is a \(2 \times 2\) matrix, which indicates it has 2 rows and 2 columns creating a square matrix. Another matrix, \(B\), is a \(2 \times 3\) matrix, having 2 rows and 3 columns. Recognizing these dimensions intuitively helps ensure that you can apply the matrices properly in operations such as multiplication, addition, and more complex algebraic manipulations.

Compatibility for Multiplication

Checking for compatibility is crucial when multiplying matrices. Not all matrices can be multiplied with each other. The rule is simple: the number of columns in the first matrix must match the number of rows in the second matrix for the multiplication to be possible.

For example, consider matrices \(D\) and \(A\) from the original exercise. Matrix \(D\) has dimensions \(1 \times 2\), and matrix \(A\) is \(2 \times 2\). Since the number of columns in \(D\) (which is 2) matches the number of rows in \(A\) (also 2), these matrices are compatible for multiplication. The resulting product of matrices \(D\) and \(A\) is a matrix of dimensions \(1 \times 2\).

• If the inner dimensions do not match, multiplication is not possible.
• The outer dimensions determine the size of the resulting matrix.

Be sure to always verify compatibility before executing the multiplication.

Matrix Algebra

Matrix algebra includes the study and application of operations on matrices. It extends many of the concepts of algebra to matrices, allowing complex computations and transformations.

In matrix algebra, you perform operations like addition, subtraction, and multiplication. Addition and subtraction require that matrices be of the same dimensions. However, multiplication does not, as long as the column-row rule is obeyed. Furthermore, the order of factors in matrix multiplication matters; changing the order usually results in a different product or even makes the operation impossible. This property makes it distinct from scalar algebra where multiplication is commutative.

Additionally, matrices can have identities, like the identity matrix \(F\) shown in the exercise with values aligned along the diagonal. This identity property acts as the multiplicative counterpart to 1 in real numbers, where multiplying any matrix by an identity matrix of compatible size results in the original matrix.

Matrix Operations

Matrix operations are the procedures or rules applied to matrices that allow computation between them. A common operation is multiplication, which allows matrices to transform vectors or other matrices in complex ways.

When performing the multiplication of two matrices, you systematically multiply and sum combinations of elements across rows and columns according to specific rules:

• Take the elements of the first matrix's row and the corresponding elements of the second matrix's column.
• Multiply them together – element by element.
• Sum the results to get each entry in the new matrix.

In the example, multiplying \((DA)B\) involves taking the resultant \(1 \times 2\) matrix \(DA\) and multiplying it with matrix \(B\), a \(2 \times 3\) matrix. This results in a final \(1 \times 3\) matrix, as seen in the solution.

Mastering matrix operations broadens your understanding of not just mathematics but its real-world application in systems such as graphics, physics simulations, and more.