Problem 18 - College Algebra

Question

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

Step-by-Step Solution

Answer

The product matrix \(AB\) is \(\begin{bmatrix}-43 & -6 & 56 \\ -7 & 18 & 36 \end{bmatrix}\).

Step 1: Introduction to Matrix Multiplication

To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Here, matrix \(A\) is a \(2 \times 2\) matrix, and matrix \(B\) is a \(2 \times 3\) matrix, making the multiplication possible. The result will be a \(2 \times 3\) matrix.

Step 2: Multiply First Row of A by Each Column of B

To find the element in the first row and first column of the resulting matrix, multiply the first row of matrix \(A\) by the first column of matrix \(B\): \((-1) \cdot 3 + 5 \cdot (-8) = -3 - 40 = -43\) For the first row, second column: \((-1) \cdot 6 + 5 \cdot 0 = -6 + 0 = -6\) For the first row, third column: \((-1) \cdot 4 + 5 \cdot 12 = -4 + 60 = 56\) Thus, the first row of the result is \([-43, -6, 56]\).

Step 3: Multiply Second Row of A by Each Column of B

To find the element in the second row and first column of the resulting matrix, multiply the second row of matrix \(A\) by the first column of matrix \(B\): \(3 \cdot 3 + 2 \cdot (-8) = 9 - 16 = -7\) For the second row, second column: \(3 \cdot 6 + 2 \cdot 0 = 18 + 0 = 18\) For the second row, third column: \(3 \cdot 4 + 2 \cdot 12 = 12 + 24 = 36\) Thus, the second row of the result is \([-7, 18, 36]\).

Step 4: Assemble the Resulting Matrix

Combine the results from Steps 2 and 3 to form the final product matrix:\[A B = \begin{bmatrix}-43 & -6 & 56 \ -7 & 18 & 36 \end{bmatrix}\]

Key Concepts

Matrix Operations

Matrix operations involve manipulations or calculations that you can perform on matrices. These operations include addition, subtraction, and multiplication, among others. Each operation has its own specific rules and requirements. Matrix multiplication, for example, is not like regular number multiplication. It's more complex and involves a row-by-column approach. When multiplying matrices, you follow a step-by-step process to ensure each element of the resultant matrix is calculated correctly. Each element in the resultant matrix comes from multiplying elements from rows of the first matrix and columns of the second matrix. Understanding matrix operations is crucial in many areas of mathematics and engineering, as they are foundational in more advanced topics like linear transformations and vector spaces.

Matrix Dimensions

Matrix dimensions are described in terms of their rows and columns, such as "2 x 3" for a matrix with 2 rows and 3 columns. Knowing the dimensions of a matrix is vital, particularly when performing matrix operations such as multiplication. For multiplication, the number of columns in the first matrix must match the number of rows in the second. If the dimensions are compatible, matrix multiplication is possible. In this exercise, matrix A is a 2x2 matrix, and matrix B is a 2x3 matrix, which makes the matrix multiplication possible, since A's columns (2) match B's rows (2). The resulting matrix of multiplication will have dimensions determined by the rows of the first matrix and the columns of the second matrix. Here, the result is a 2x3 matrix because A has 2 rows and B has 3 columns.

Resultant Matrix

The resultant matrix is the product of multiplying two matrices together. When you multiply matrices A and B, you create a new matrix with its elements derived from specific calculations. To find each element in the resultant matrix, you multiply the elements of a particular row of the first matrix by the corresponding columns of the second matrix. Then, you add up these products. In our example, the first element of the first row in the resulting matrix is calculated as \((-1) \cdot 3 + 5 \cdot (-8) = -43\). You repeat these steps for every element in the new matrix. The resultant matrix for the multiplication of matrices A and B is: \[\begin{bmatrix}-43 & -6 & 56 \ -7 & 18 & 36 \end{bmatrix}\].

Matrices A and B

Matrices A and B are the initial matrices chosen for the matrix multiplication process in this exercise. They have specific values and dimensions that determine the steps used to achieve the final product matrix. Matrix A is a 2x2 matrix given by \[A=\begin{bmatrix}-1 & 5 \ 3 & 2 \end{bmatrix}\], and matrix B is a 2x3 matrix given by \[B=\begin{bmatrix}3 & 6 & 4 \ -8 & 0 & 12 \end{bmatrix}\]. These matrices are selected based on their compatible dimensions, which allow matrix multiplication. Working with matrices A and B helps illustrate the rules and challenges involved in matrix operations, making it easier to learn about matrix multiplication.