Problem 29 - College Algebra and Calculus: An Applied Approach

Question

Find \(A B\), if possible. $$ A=\left[\begin{array}{rrr} 5 & 0 & 0 \\ 0 & -8 & 0 \\ 0 & 0 & 7 \end{array}\right], B=\left[\begin{array}{rrr} \frac{1}{5} & 0 & 0 \\ 0 & -\frac{1}{8} & 0 \\ 0 & 0 & \frac{1}{2} \end{array}\right] $$

Step-by-Step Solution

Answer

The resulting matrix is \(AB =\left[\begin{array}{ccc}\n1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 3.5 \n\end{array}\right]\)

Step 1: Check the Size of Each Matrix

Matrix A is a 3x3 matrix, and matrix B is also a 3x3 matrix. In order to perform the matrix multiplication operation, the number of columns in the first matrix must be equal to the number of rows in the second matrix, which is the case here.

Step 2: Perform the Multiplication

Multiplication in matrices is done element by element. Meaning that element in row i and column j of the resultant matrix is obtained by multiplying element of row i in matrix A by element of column j in matrix B. Thus, the product AB gives the matrix:\n \(AB =\left[\begin{array}{ccc}\n5*1/5 & 0*0 & 0*0 \\ 0*0 & -8*-1/8 & 0*0 \\ 0*0 & 0*0 & 7*1/2 \n\end{array}\right]\) Which simplifies to: \(AB =\left[\begin{array}{ccc}\n1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 3.5 \n\end{array}\right]\)

Step 3: Present the Result

The resulting matrix \(AB\) is a 3x3 matrix with 1 in the first row, first column and 1 in the second row, second column, and 3.5 in the third row, third column. This matrix is a diagonal matrix, where only the diagonal contains non-zero values.

Key Concepts

Diagonal Matrix

A diagonal matrix is a type of square matrix where all the elements outside the main diagonal are zero. This means that only the values along the diagonal, from the top-left to the bottom-right of the matrix, have non-zero values, if any at all. This structure makes certain mathematical operations, like matrix multiplication, notably simpler.
Diagonal matrices have special properties that make them quite useful:

• They are easy to invert, assuming no diagonal element is zero, by simply taking the reciprocal of each non-zero diagonal element.
• Multiplying two diagonal matrices together is straightforward: just multiply their corresponding diagonal elements.
• Any power of a diagonal matrix is another diagonal matrix, where each diagonal element is raised to the power.

For example, in our exercise, both matrices \( A \) and \( B \) are diagonal. Notice how the non-diagonal elements don't influence the matrix multiplication result because they are zero.

Matrix Operations

Matrix operations include addition, subtraction, multiplication, and more advanced operations like finding determinants and inverses. Matrix multiplication is especially significant because it has distinct rules compared to basic arithmetic.

When multiplying matrices, each element is a sum of products from the rows of the first matrix and the columns of the second. This means:

• The number of columns in the first matrix must equal the number of rows in the second matrix.
• Every element \((i, j)\) in the resulting matrix is obtained by multiplying each element of the \(i\)-th row of the first matrix with the corresponding element of the \(j\)-th column of the second matrix, and then summing the products.
• The order of multiplication matters: \( A \times B eq B \times A \) in general.

In our exercise, matrix \( A \) and matrix \( B \) are both 3x3 matrices, which makes them compatible for matrix multiplication. Since they are diagonal, the result is straightforward and simplifies the multiplication process.

3x3 Matrix

A 3x3 matrix is a square matrix composed of three rows and three columns, leading to nine elements in total.
This type of matrix is common in various areas of mathematics and physics, such as transformations in 3D space.

• Being a square matrix, it allows for operations like determinant calculation and easy implementation of diagonal matrices.
• 3x3 matrices often serve as a simple yet powerful tool for modeling systems and are widely used in linear algebra.
• When working with 3x3 matrices, checking compatibility for multiplication or other operations is quite manageable since the dimensions are small.

In the given exercise, both matrices \( A \) and \( B \) are 3x3 matrices. They represent simple cases for understanding the essential operations and properties of matrices.