Problem 37 - Precalculus with Limits

Question

In Exercises 33-40, if possible, find \(AB\) and state the order of the result. \(A=\left[\begin{array}{r} 5 & 0 & 0 \\ 0 & -8 & 0 \\ 0 & 0 & 7 \end{array}\right]\), \(B=\left[\begin{array}{r} \frac{1}{5} & 0 & 0 \\ 0 & -\frac{1}{8} & 0 \\ 0 & 0 & \frac{1}{2} \end{array}\right]\)

Step-by-Step Solution

Answer

The product of matrices A and B is a 3x3 matrix: \(\[\begin{array}{ccc} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & \frac{7}{2} \end{array}\]\).

Step 1: Understand Matrix Multiplication

Matrix multiplication involves taking the dot product of the rows of the first matrix with the columns of the second matrix.

Step 2: Apply the Formula

Multiply the corresponding entries of the matrices. Since both matrices are diagonal, their product results in another diagonal matrix where each diagonal entry is the product of corresponding entries of A and B.

Step 3: Carry out the Multiplication

Perform the operation: \(AB\) = \[\begin{array}{ccc} 5*\frac{1}{5} & 0 & 0 \ 0 & -8*-\frac{1}{8} & 0 \ 0 & 0 & 7* \frac{1}{2} \end{array}\] = \[\begin{array}{ccc} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & \frac{7}{2} \end{array}\]

Key Concepts

Diagonal Matrix

A diagonal matrix is a special form of square matrix where all the elements outside the main diagonal are zero. The main diagonal is the diagonal that runs from the top left to the bottom right. Diagonal matrices are denoted by the symbol \(D\). For instance, the matrix \(A\) given in the original exercise is a diagonal matrix:

• \(A = \begin{bmatrix} 5 & 0 & 0 \ 0 & -8 & 0 \ 0 & 0 & 7 \end{bmatrix}\)

This means that only the values 5, -8, and 7 are non-zero. Diagonal matrices simplify many mathematical operations like matrix multiplication, inversion, and finding eigenvalues. In terms of multiplication, if both matrices are diagonal, as in our solution, the resulting matrix is also diagonal. This property greatly simplifies calculations, as you only need to multiply the corresponding diagonal elements to find each element of the resulting matrix.

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in linear algebra. In the context of matrix multiplication, the dot product is used to multiply the elements of matrices to produce a new matrix. To perform a dot product using matrices:

• Take a row from the first matrix.
• Multiply it by a column from the second matrix.
• The result is a single number, which forms an element of the resulting matrix.

Since both matrices \( A \) and \( B \) are diagonal, their multiplication is straightforward. The dot product involves only multiplying corresponding diagonal elements:

• \(5 \times \frac{1}{5} = 1\)
• \(-8 \times -\frac{1}{8} = 1\)
• \(7 \times \frac{1}{2} = \frac{7}{2}\)

This property of diagonal matrix multiplication makes it easier and quicker to compute the product, as there's no need for complex row and column matching across non-diagonal elements.

Matrix Product Order

The order of the matrix product is an important aspect of matrix multiplication. The order refers to the dimensions (rows and columns) of a matrix. For example, if matrix \(A\) has dimensions \(m \times n\) and matrix \(B\) has dimensions \(n \times p\), their product \(AB\) is a matrix of order \(m \times p\).

• For matrix \(A\) and \(B\) in our exercise, both matrices are of size \(3 \times 3\).

When you multiply two \(3 \times 3\) matrices, the result is also a \(3 \times 3\) matrix. This size is derived from taking the number of rows of the first matrix and the number of columns of the second matrix. Understanding the matrix product order is crucial as it ensures that the dimensions align properly for the multiplication to be valid and result in a correctly sized matrix.