Question

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

Short answer

The products are \( D A = \left[\begin{array}{cc} d_1 a & d_1 b \\ d_2 c & d_2 d \end{array}\right] \) and \( A D = \left[\begin{array}{cc} a d_1 & b d_2 \\ c d_1 & d d_2 \end{array}\right] \).

Step-by-step solution

Understand Matrix Multiplication

The product of two matrices is found by taking the dot product of the rows of the first matrix with the columns of the second matrix. The resulting matrix has dimensions matching the number of rows of the first matrix and columns of the second matrix.

Multiplying Diagonal Matrix D by Matrix A (D A)

To compute the product \( D A \), multiply each row in \( D \) by each column in \( A \). Since \( D \) is a diagonal matrix, the multiplication simplifies as follows: \[ D A = \left[\begin{array}{cc} d_1 & 0 \ 0 & d_2 \end{array}\right] \left[\begin{array}{cc} a & b \ c & d \end{array}\right] = \left[\begin{array}{cc} d_1 \, a & d_1 \, b \ d_2 \, c & d_2 \, d \end{array}\right]. \]

Verify Dimensions for Possible Multiplication A D

Before attempting to multiply \( A \) by \( D \), ensure the dimensions align correctly. For multiplication \( A D \), the number of columns in \( A \) must equal the number of rows in \( D \). In this case, both are 2x2 matrices, so multiplication is possible.

Multiplying Matrix A by Diagonal Matrix D (A D)

Now compute \( A D \) using matrix multiplication: \[ A D = \left[\begin{array}{cc} a & b \ c & d \end{array}\right] \left[\begin{array}{cc} d_1 & 0 \ 0 & d_2 \end{array}\right] = \left[\begin{array}{cc} a \, d_1 & b \, d_2 \ c \, d_1 & d \, d_2 \end{array}\right]. \]

Compare Results

After calculating both multiplication scenarios, compare the resulting matrices: \( D A = \left[\begin{array}{cc} d_1 a & d_1 b \ d_2 c & d_2 d \end{array}\right] \) and \( A D = \left[\begin{array}{cc} a d_1 & b d_2 \ c d_1 & d d_2 \end{array}\right] \). These matrices are generally different, showing that matrix multiplication is not commutative.

Key concepts

Diagonal Matrix

In the world of linear algebra, a diagonal matrix is a special kind of square matrix where all the elements outside the main diagonal are zero. The main diagonal itself can consist of any values and is where the identity of the diagonal matrix is derived. This unique structure allows diagonal matrices to simplify certain operations like matrix multiplication.

For example, consider a 2x2 diagonal matrix:

• The form is: \( D = \begin{bmatrix} d_1 & 0 \ 0 & d_2 \end{bmatrix} \).
• The non-zero elements, \(d_1\) and \(d_2\), exist only on the main diagonal.
• Off-diagonal elements are zero, making calculations like multiplication more straightforward.

This simplification arises because multiplying a diagonal matrix by another matrix affects only specific rows or columns, based on non-zero entries. It makes these matrices especially useful computationally.

Commutative Property

The commutative property is an easy-to-understand principle often associated with basic arithmetic operations like addition and multiplication. When we say two operations are commutative, swapping the operands does not change the result. For example, with numbers: \(3 + 5 = 5 + 3\).

• In matrix multiplication, however, this does not generally apply.
• Thanks to the distinct layout and dimensions of matrices, \(AB\) usually does not equal \(BA\).
• This discrepancy arises because the order of multiplication affects the rows and columns used to calculate the result.

Even with matrices like diagonal matrices, the commutative property does not hold unless both matrices share specific properties. Hence, when dealing with matrices, always check each multiplication direction individually.

Matrix Dimensions

Matrix dimensions are crucial in determining whether matrices can be multiplied. They are defined as "rows by columns," a notation that immediately tells you the shape of a matrix. Understanding these dimensions helps solve crucial questions in multiplication feasibility.

Consider a dimension outline:

• A matrix with dimensions \( m \times n \) has \( m \) rows and \( n \) columns.
• The matrices can only be multiplied if the number of columns in the first matrix equals the number of rows in the second.

For our given matrices:

• Both \(D\) and \(A\) are 2x2 matrices.
• This alignment allows for multiplication in both directions \((DA)\) and \((AD)\).

Being mindful of dimensions ensures successful matrix operations, avoiding math errors during calculations.

Dot Product

The dot product is a key concept in matrix multiplication that combines two sequences of numbers into a single number. It is the sum of the products of corresponding entries. The dot product is used to achieve each element's result in a matrix multiplication. Here's how it works:

• Take two vectors: \([a, b]\) and \([c, d]\).
• The dot product would be \( a \cdot c + b \cdot d \).

In matrix multiplication:

• Rows of the first matrix are paired element-wise with columns of the second.
• The results form each element of the product matrix.

For example:
Multiplying matrices \( D = \begin{bmatrix} d_1 & 0 \ 0 & d_2 \end{bmatrix} \) and \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), we calculate each element of the resulting matrix using the dot products of rows from \(D\) and columns from \(A\).

This process brings depth to the concept of matrix multiplications, turning it into a structured yet dynamic process for linear transformations.