Problem 21 - College Algebra

Question

Use the matrices below to perform matrix multiplication. \(A=\left[\begin{array}{rr}-1 & 5 \\ 3 & 2\end{array}\right], B=\left[\begin{array}{rrr}3 & 6 & 4 \\ -8 & 0 & 12\end{array}\right], C=\left[\begin{array}{rr}4 & 10 \\ -2 & 6 \\ 5 & 9\end{array}\right], D=\left[\begin{array}{rrr}2 & -3 & 12 \\ 9 & 3 & 1 \\ 0 & 8 & -10\end{array}\right]\) \(B D\)

Step-by-Step Solution

Answer

The resultant matrix of \(BD\) is \(\begin{bmatrix} 60 & 41 & 2 \\ -16 & 120 & -216 \end{bmatrix}\).

Step 1: Verify if Multiplication is Possible

Check the dimensions of matrices \(B\) and \(D\). Matrix \(B\) has dimensions \(2 \times 3\) and matrix \(D\) has dimensions \(3 \times 3\). Since the number of columns in \(B\) matches the number of rows in \(D\), matrix multiplication \(BD\) is possible.

Step 2: Set Up Matrix Multiplication

The resulting matrix \(BD\) will have dimensions \(2 \times 3\). We will calculate each element by taking the dot product of rows from matrix \(B\) with columns from matrix \(D\).

Step 3: Calculate Each Element of Resulting Matrix

Let's calculate the elements of matrix \(BD\):Element (1,1): \((3)(2) + (6)(9) + (4)(0) = 6 + 54 + 0 = 60\)Element (1,2):\((3)(-3) + (6)(3) + (4)(8) = -9 + 18 + 32 = 41\)Element (1,3):\((3)(12) + (6)(1) + (4)(-10) = 36 + 6 - 40 = 2\)Element (2,1):\((-8)(2) + (0)(9) + (12)(0) = -16 + 0 + 0 = -16\)Element (2,2):\((-8)(-3) + (0)(3) + (12)(8) = 24 + 0 + 96 = 120\)Element (2,3):\((-8)(12) + (0)(1) + (12)(-10) = -96 + 0 - 120 = -216\)

Step 4: Construct the Resultant Matrix

Using the individual elements calculated, construct the resulting matrix \(BD\):\[BD = \begin{bmatrix} 60 & 41 & 2 \ -16 & 120 & -216 \end{bmatrix}\]

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is key in determining whether matrix multiplication is possible. The dimensions of a matrix are described as rows by columns. For instance, matrix \(B\) is a 2x3 matrix, which means it has 2 rows and 3 columns. Similarly, matrix \(D\) is 3x3, having 3 rows and 3 columns.
The rule for matrix multiplication is that the number of columns in the first matrix must match the number of rows in the second matrix. This means that for two matrices to be multiplied, such as matrix \(B\) and matrix \(D\), the column count of \(B\) (3 columns) must equal the row count of \(D\) (3 rows), which it does. Hence, these matrices can be multiplied.
It's essential to memorize this rule, as understanding matrix dimensions helps in both the theoretical and practical application of matrix multiplication.

Dot Product

In matrix multiplication, the dot product plays a crucial role in determining the individual elements of the resultant matrix. A dot product is calculated by multiplying corresponding elements of two sequences and summing those products.
When multiplying two matrices, each element in the resulting matrix is a result of the dot product of a row from the first matrix with a column from the second matrix. For example, the dot product for the first element in the result matrix \(BD\) is calculated by taking the first row of matrix \(B\) and the first column of matrix \(D\).
Being proficient in calculating dot products ensures accuracy in constructing the resultant matrix, as each element is a summation of multiple calculated values.

Resultant Matrix

The resultant matrix is the product of the matrix multiplication process. When matrices \(B\) and \(D\) are multiplied, the result is a new matrix, \(BD\), whose dimensions are determined by the row count of the first matrix and the column count of the second matrix.
For the multiplication of \(B\) (2x3) and \(D\) (3x3), the dimensions of the resultant matrix \(BD\) will be 2x3, since it takes the row count from \(B\) and the column count from \(D\).
The resultant matrix presents all calculated elements derived from the dot products of the rows and columns of the multiplying matrices. It indicates the transformed output based on the input matrices and the rules of matrix multiplication.

Matrix Element Calculation

Calculating individual elements of a resultant matrix involves applying the dot product technique. Let's break down this process with an example from matrices \(B\) and \(D\).
To calculate an element in the first row and first column of the resulting matrix \(BD\), we take the first row of \(B\) and the first column of \(D\): \((3\times2) + (6\times9) + (4\times0) = 6 + 54 + 0 = 60\). This is the element located at position (1,1).
Continuing this method for each position, you need to ensure each computation is performed carefully. This involves iterating over each row of the first matrix and each column of the second matrix until all elements in the resultant matrix are calculated.
Achieving accurate matrix element calculations is fundamental in matrix algebra, as this directly influences the correctness of the entire matrix outcome.