Problem 30 - 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 product \( D(AB) \) is \([28, 21, 28]\).

Step 1: Check Matrix Multiplicability

To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Here, we need to check if matrices \( A \) and \( B \) can be multiplied. Matrix \( A \) is of size \(2 \times 2\) and matrix \( B \) is of size \(2 \times 3\). Since the number of columns in \( A (2) \) equals the number of rows in \( B (2) \), the matrices can be multiplied.

Step 2: Multiply Matrices A and B

Multiply matrices \( A \) and \( B \) by calculating the dot product of the rows of \( A \) with the columns of \( B \). The resulting matrix \( AB \) will be of size \(2 \times 3\). The elements are calculated as follows:\[AB = \begin{bmatrix}2 \times 3 + (-5) \times 1 & 2 \times \frac{1}{2} + (-5) \times (-1) & 2 \times 5 + (-5) \times 3 \0 \times 3 + 7 \times 1 & 0 \times \frac{1}{2} + 7 \times (-1) & 0 \times 5 + 7 \times 3\end{bmatrix} =\begin{bmatrix}1 & 6 & -5 \7 & -7 & 21\end{bmatrix}\]

Step 3: Check Multiplicability of D and Resulting Matrix AB

Now, we need to check if matrix \( D \), which is of size \(1 \times 2\), can be multiplied with matrix \( AB \), which is of size \(2 \times 3\). The number of columns in \( D \) (2) matches the number of rows in \( AB \) (2), so the multiplication \( D(AB) \) is possible.

Step 4: Multiply Matrices D and AB

Multiply matrix \( D \) by matrix \( AB \) by calculating the dot product of the row of \( D \) with the columns of \( AB \). The resulting matrix \( D(AB) \) will have dimensions \(1 \times 3\) as shown: \[D(AB) = \begin{bmatrix} 7 & 3 \end{bmatrix} \times \begin{bmatrix} 1 & 6 & -5 \ 7 & -7 & 21 \end{bmatrix} = \begin{bmatrix} 7\times 1 + 3\times 7 & 7\times 6 + 3\times (-7) & 7\times (-5) + 3\times 21 \end{bmatrix}\]Calculating it gives the final matrix:\[D(AB) = \begin{bmatrix} 28 & 21 & 28 \end{bmatrix}\]

Key Concepts

Dot Product

The dot product is a fundamental concept in matrix multiplication. It's a way to combine two vectors into a single scalar. Here's how it works in matrix multiplication: for each element in the resulting matrix, you multiply the elements of a row in the first matrix by the corresponding elements of a column in the second matrix, and then sum up those products.
For instance, when you compute the element in the first row and first column of the product of matrices A and B, you multiply each element of the first row of A by each corresponding element of the first column of B and add them together. In mathematical terms, for two vectors \(\mathbf{a}\) and \(\mathbf{b}\), the dot product is defined as: \[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + \dots + a_nb_n \]In matrix terms, this operation is repeated for each element of the resulting matrix.

Matrix Dimensions

Matrix dimensions describe the size of a matrix through specifying the number of rows and columns it contains. Understanding matrix dimensions is critical for matrix multiplication, as it determines the 'shape' of the matrix. The dimensions of a matrix are usually written as 'rows x columns'.
For example, if you have a matrix of dimensions 2 x 3, it means that the matrix has 2 rows and 3 columns.
The matrix dimensions are crucial because they affect how you can multiply matrices together. For two matrices to be multiplied, the number of columns in the first matrix must match the number of rows in the second matrix. If the dimensions are compatible, you'll end up with a new matrix whose dimensions come from the outer dimensions of the multiplied matrices.

Matrix Algebra

Matrix algebra is the set of rules and processes used to perform operations on matrices, such as addition, subtraction, and multiplication. Matrix algebra extends regular algebra to a system that can handle arrays of numbers.
In this type of algebra, matrices are typically used to perform linear transformations or solve systems of linear equations. Matrix multiplication is one of the most common operations in matrix algebra. Unlike regular multiplication, matrix multiplication combines two matrices to result in another matrix. The operation is not commutative, meaning that \(AB eq BA\).
Matrix algebra also includes properties such as the identity matrix, which acts much like the number 1 in regular multiplication. When multiplied by another matrix, it leaves that matrix unchanged.

Multiplier Compatibility

Multiplier compatibility refers to the requirement for matrix dimensions to align properly for multiplication to occur. Specifically, the number of columns in the first matrix must equal the number of rows in the second matrix.
This compatibility ensures that you can perform the dot product between rows of the first matrix and columns of the second matrix.
For instance, if you have a 2 x 3 matrix and want to multiply it by a 3 x 4 matrix, you can do so because the inner dimensions match (3 columns in the first matrix and 3 rows in the second). The resulting matrix will have the outer dimensions, which is 2 x 4 in this case.
Understanding this compatibility is essential when working with matrices because it dictates whether matrix multiplication is possible.