Problem 38 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The product of the given matrices A and B is a matrix\[C = \left[\begin{array}{ccc}1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 3.5\end{array}\right]\]

Step 1: Analyze the Matrices

The matrices given, A and B, are both 3x3 matrices. It's crucial to note that these are square matrices and also diagonal matrices. A diagonal matrix is a matrix where elements outside the main diagonal are all zero.

Step 2: Set the Resultant Matrix

We are to find the product of A and B written as AB. The elements of the resultant matrix are obtained by multiplying the corresponding elements of the given matrices because they are diagonal.

Step 3: Compute the Elements of the Resultant Matrix

We perform element-wise multiplication across A and B. That is, for each element a_{ij} in A, we multiply by b_{ij} in B to get the corresponding element c_{ij} in the resultant matrix C. Doing this gives:\[C = \left[\begin{array}{ccc}5*\frac{1}{5} & 0*0 & 0*0 \0*0 & -8 * -\frac{1}{8} & 0 * 0 \0 * 0 & 0 * 0 &7*\frac{1}{2} \end{array}\right]\]After calculating, we get:\[C = \left[\begin{array}{ccc}1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 3.5\end{array}\right]\]

Key Concepts

Diagonal Matrix

A diagonal matrix is a special type of square matrix where all off-diagonal elements are zero. This means that only the elements on the main diagonal, which runs from the top left corner to the bottom right corner, can be non-zero.
In the context of the exercise, both matrices A and B are diagonal matrices. This property makes calculations, such as matrix multiplication, much simpler.

• Reduced Complexity: Multiplying diagonal matrices is straightforward since you only need to multiply corresponding diagonal elements.
• Simplicity: Outside the diagonal, all positions have zero values, which simplifies computations and makes it easy to identify.

Understand that while diagonal matrices simplify many matrix operations, they do have limitations, such as being unable to represent more complex transformations.

Square Matrix

A square matrix is defined as a matrix with the same number of rows and columns. For example, a 3x3 matrix has three rows and three columns.
Both matrices A and B in the exercise are square matrices. This form is significant because many properties and operations in linear algebra, such as determinants and inverses, are only defined for square matrices.

• Consistency: The number of elements in each row is the same as the number of elements in each column.
• Key Properties: Only square matrices can potentially be invertible, have eigenvalues, and be diagonalizable.

Square matrices frequently appear in mathematical problems, particularly those involving transformations and linear equations.

Element-wise Multiplication

Element-wise multiplication is a unique operation that applies primarily when working with diagonal matrices. For the matrices in this exercise, it means multiplying corresponding elements from the two matrices.
This is different from the usual matrix multiplication, where rows and columns are used in dot product calculations.

• Simplified Multiplication: Since only non-zero elements are on the diagonal, you multiply only these positions directly.
• Efficiency: This reduces the computational workload as zero-valued positions do not contribute to the operation.

In the product matrix C from the exercise, each element on the diagonal of matrix C comes from multiplying corresponding diagonal elements of matrices A and B: 5 times \(\frac{1}{5}\), -8 times \(-\frac{1}{8}\), and 7 times \(\frac{1}{2}\). This approach highlights how diagonal matrices make it straightforward to use element-wise multiplication rather than the traditional matrix operation.