Question

A diagonal matrix \(D\) and a matrix \(A\) are given. Find the products \(D A\) and \(A D,\) where possible. $$ \begin{array}{l} D=\left[\begin{array}{ccc} 1 & 1 & 1 \\ 2 & 2 & 2 \\ -3 & -3 & -3 \end{array}\right] \\ A=\left[\begin{array}{ccc} 2 & 0 & 0 \\ 0 & -3 & 0 \\ 0 & 0 & 5 \end{array}\right] \end{array} $$

Short answer

DA = \( \begin{bmatrix} 2 & -3 & 5 \\ 4 & -6 & 10 \\ -6 & 9 & -15 \end{bmatrix} \), and AD = \( \begin{bmatrix} 2 & 2 & 2 \\ -6 & -6 & -6 \\ -15 & -15 & -15 \end{bmatrix} \).

Step-by-step solution

Define Matrix Multiplication

To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Given matrices: \( D \) is a \(3 \times 3\) matrix and \( A \) is a \(3 \times 3 \) diagonal matrix. Thus, both \( D A \) and \( A D \) are possible as both will result in \(3 \times 3\) matrices.

Calculate DA

Perform the multiplication \( D A \). Since \( A \) is a diagonal matrix, the diagonal elements will scale the corresponding rows of \( D \).\[D A = \begin{bmatrix} 1 & 1 & 1 \ 2 & 2 & 2 \ -3 & -3 & -3 \end{bmatrix} \begin{bmatrix} 2 & 0 & 0 \ 0 & -3 & 0 \ 0 & 0 & 5 \end{bmatrix}\]Multiply row by column:\[D A = \begin{bmatrix} 1\times2 + 1\times0 + 1\times0 & 1\times0 + 1\times-3 + 1\times0 & 1\times0 + 1\times0 + 1\times5 \ 2\times2 + 2\times0 + 2\times0 & 2\times0 + 2\times-3 + 2\times0 & 2\times0 + 2\times0 + 2\times5 \ -3\times2 + -3\times0 + -3\times0 & -3\times0 + -3\times-3 + -3\times0 & -3\times0 + -3\times0 + -3\times5 \end{bmatrix}\]Resulting in:\[D A = \begin{bmatrix} 2 & -3 & 5 \ 4 & -6 & 10 \ -6 & 9 & -15 \end{bmatrix}\]

Calculate AD

Multiply \( A \) and \( D \). Each element of a row in \( A \) will be multiplied by the corresponding column in \( D \).\[A D = \begin{bmatrix} 2 & 0 & 0 \ 0 & -3 & 0 \ 0 & 0 & 5 \end{bmatrix} \begin{bmatrix} 1 & 1 & 1 \ 2 & 2 & 2 \ -3 & -3 & -3 \end{bmatrix}\]Multiply row by column:\[A D = \begin{bmatrix} 2\times1 + 0\times2 + 0\times-3 & 2\times1 + 0\times2 + 0\times-3 & 2\times1 + 0\times2 + 0\times-3 \ 0\times1 + -3\times2 + 0\times-3 & 0\times1 + -3\times2 + 0\times-3 & 0\times1 + -3\times2 + 0\times-3 \ 0\times1 + 0\times2 + 5\times-3 & 0\times1 + 0\times2 + 5\times-3 & 0\times1 + 0\times2 + 5\times-3 \end{bmatrix}\]Resulting in:\[A D = \begin{bmatrix} 2 & 2 & 2 \ -6 & -6 & -6 \ -15 & -15 & -15 \end{bmatrix}\]

Key concepts

Diagonal Matrix

A diagonal matrix is a special kind of matrix where all the elements outside the main diagonal are zero. The main diagonal is the one that spans from the top left to the bottom right corner. For example, in matrix \( A \) given in the exercise:

• Top-left to bottom-right: The elements are \( 2, -3, \) and \( 5 \).
• All other entries are zeros.

Diagonal matrices are simple yet powerful for calculations. They make some operations, like matrix multiplication, easier and faster. In particular, when you multiply any matrix by a diagonal matrix, you essentially scale the rows or columns of the other matrix by the diagonal entries. This property can simplify many calculations in linear algebra.

Matrix Product

Matrix product or matrix multiplication involves combining two matrices to form a new matrix. To multiply two matrices:

• The number of columns in the first matrix must match the number of rows in the second matrix.
• Here, both matrices \( D \) and \( A \) are 3x3, allowing for multiplication in both orders: \( DA \) and \( AD \).

In this process, each element in the resulting matrix is the sum of the products of elements from the corresponding row of the first matrix and the column of the second matrix. Thus, each element requires a combination of these row-by-column products.

In the exercise, multiplying diagonals, like in \( A \), with any matrix really simplifies the computation.

• For \( DA \), each row of \( D \) is multiplied by the corresponding diagonal entry of \( A \).
• For \( AD \), each column of \( D \) is repeated along the rows, scaled by the diagonal elements of \( A \).

Matrices

Matrices are made up of rows and columns. Think of them as a grid where each number sits in a specific position. Understanding matrices is key in solving linear equations, transforming data, and many practical applications in science and engineering.

Here are a few foundational aspects of matrices:

• A matrix's size, or order, is defined by its number of rows and columns. For example, \( D \) and \( A \) are both 3x3 matrices in this exercise, meaning they each have 3 rows and 3 columns.
• The notation for a matrix typically uses brackets, and each element is usually indexed as \( a_{ij} \), where \( i \) is the row number and \( j \) is the column number.
• Adding and subtracting matrices requires them to be of the same size, while multiplication is more flexible as explained in the matrix product section.

In practical terms, matrices help organize information, like equations and data sets, in ways that make calculations more manageable. By grasping the basics of matrices, you build a foundation for exploring more intricate concepts in linear algebra.