Question

A diagonal matrix \(D\) and a matrix \(A\) are given. Find the products \(D A\) and \(A D,\) where possible. $$ \begin{array}{l} D=\left[\begin{array}{ccc} d_{1} & 0 & 0 \\ 0 & d_{2} & 0 \\ 0 & 0 & d_{3} \end{array}\right] \\ A=\left[\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & i \end{array}\right] \end{array} $$

Short answer

The products are \( DA = \begin{bmatrix} d_1a & d_1b & d_1c \\ d_2d & d_2e & d_2f \\ d_3g & d_3h & d_3i \end{bmatrix} \) and \( AD = \begin{bmatrix} d_1a & d_2b & d_3c \\ d_1d & d_2e & d_3f \\ d_1g & d_2h & d_3i \end{bmatrix} \).

Step-by-step solution

Understand Matrix Multiplication

Matrix multiplication can be done only if the number of columns in the first matrix is equal to the number of rows in the second matrix. Both given matrices are 3x3, so the multiplication of both \( DA \) and \( AD \) is possible.

Compute the Product DA

To find \( DA \), multiply the diagonal matrix \( D \) by matrix \( A \). Recall that a diagonal matrix \( D \) only scales the rows of \( A \) by the diagonal elements of \( D \). Set up the multiplication as follows: \[ \begin{bmatrix} d_1 & 0 & 0 \ 0 & d_2 & 0 \ 0 & 0 & d_3 \end{bmatrix} \times \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} = \begin{bmatrix} d_1a & d_1b & d_1c \ d_2d & d_2e & d_2f \ d_3g & d_3h & d_3i \end{bmatrix} \].

Compute the Product AD

To find \( AD \), multiply matrix \( A \) by the diagonal matrix \( D \). In this process, each column of the matrix \( A \) is scaled by the corresponding diagonal element of \( D \). Set up the multiplication as follows: \[ \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \times \begin{bmatrix} d_1 & 0 & 0 \ 0 & d_2 & 0 \ 0 & 0 & d_3 \end{bmatrix} = \begin{bmatrix} d_1a & d_2b & d_3c \ d_1d & d_2e & d_3f \ d_1g & d_2h & d_3i \end{bmatrix} \].

Verify the Results

Ensure that the resulting matrices from steps 2 and 3 conform to correct matrix multiplication rules. Since \( D \) is diagonal, each row multiplication in \( DA \) corresponds to scaling by diagonal values, and each column multiplication in \( AD \) follows the same process, confirming the results.

Key concepts

Diagonal Matrix

A diagonal matrix is a special kind of square matrix. It has non-zero elements only on its main diagonal, which runs from the top left to the bottom right. The rest of the elements in the matrix are zeros.

In mathematical terms, a matrix \( D \) is diagonal if \( D = [d_{ij}] \) where \( d_{ij} = 0 \) whenever \( i eq j \). Consider our example matrix:

• \( D = \begin{bmatrix} d_1 & 0 & 0 \ 0 & d_2 & 0 \ 0 & 0 & d_3 \end{bmatrix} \)
• The only numbers that aren't zero are \( d_1, d_2, \) and \( d_3 \). These are on the diagonal.

Diagonal matrices can make certain operations, like matrix multiplication, easier because they simplify calculations by zeroing out activities that involve the off-diagonal elements.

Matrix Product

The matrix product involves multiplying two matrices together to yield a new matrix. However, you can perform multiplication only if the number of columns in the first matrix equals the number of rows in the second one. With diagonal matrices involved, the product results can simplify significantly.

For our matrices, both \( D \) and \( A \) are 3x3 matrices, meaning either \( DA \) or \( AD \) is defined:

When multiplying:

• Matrix \( DA \) scales the rows of \( A \) by the diagonal elements of \( D \).
• Matrix \( AD \) scales the columns of \( A \) by the diagonal elements of \( D \).

Matrix multiplication is not commutative, meaning \( AB \) is not necessarily the same as \( BA \). Thus, even though \( D \) and \( A \) can be multiplied in two different orders, the results are distinct and must be computed separately.

Matrix Scaling

Matrix scaling in the context of a diagonal matrix means multiplying each element by the diagonal elements. Essentially, if you have a diagonal matrix \( D \) and another matrix \( A \), the diagonal matrix will scale either the rows or columns of the second matrix depending on the multiplication order.

Here's how it works:- In \( DA \): the diagonal elements \( d_1, d_2, \) and \( d_3 \) scale the respective rows of matrix \( A \)

• \( d_1 \) scales the first row
• \( d_2 \) scales the second row
• \( d_3 \) scales the third row

- In \( AD \): the diagonal elements scale the columns of \( A \)

• \( d_1 \) scales the first column
• \( d_2 \) scales the second column
• \( d_3 \) scales the third column

Scaling with a diagonal matrix simplifies because you only change elements by multiplying them with the corresponding diagonal value.

Algebraic Operations

Algebraic operations in matrix mathematics include operations such as addition, multiplication, and subtraction. However, multiplication, especially with diagonal matrices, is particularly useful due to its practical applications in scaling, rotating, and transforming data or dimensions.

An interesting property of matrix multiplication is its distributive nature across addition, like this:\[A(B + C) = AB + AC\]This implies that you can distribute one matrix over the sum of others. Other properties like associativity and identity also play roles in more complex algebraic manipulations.

Understanding these operations is fundamental, as they allow us to represent complex systems and processes more simply. The diagonal matrix's simplifying role in such algebraic operations is vital, often making seemingly complex problems more manageable.