Question

A diagonal matrix \(D\) and a matrix \(A\) are given. Find the products \(D A\) and \(A D,\) where possible. $$ \begin{array}{l} D=\left[\begin{array}{ccc} -1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{array}\right] \\ A=\left[\begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}\right] \end{array} $$

Short answer

\(D A\) and \(A D\) are not equal; their products are different.

Step-by-step solution

Verify Compatibility for Multiplication D A

To multiply two matrices, the number of columns in the first matrix should equal the number of rows in the second matrix. Here, matrix \(D\) is a \(3 \times 3\) matrix and matrix \(A\) is also \(3 \times 3\). Since the number of columns in \(D\) equals the number of rows in \(A\), the multiplication \(D A\) is possible.

Multiply D and A

Each element of the product matrix \(D A\) is obtained by multiplying the corresponding elements of the diagonals of \(D\) with corresponding elements of \(A\). The product matrix is computed as follows:\[D A = \begin{bmatrix} -1 & 0 & 0 \0 & 2 & 0 \0 & 0 & 3 \end{bmatrix}\begin{bmatrix} 1 & 2 & 3 \4 & 5 & 6 \7 & 8 & 9 \end{bmatrix} = \begin{bmatrix} -1 \times 1 & -1 \times 2 & -1 \times 3 \2 \times 4 & 2 \times 5 & 2 \times 6 \3 \times 7 & 3 \times 8 & 3 \times 9\end{bmatrix} = \begin{bmatrix} -1 & -2 & -3 \8 & 10 & 12 \21 & 24 & 27 \end{bmatrix}.\]

Verify Compatibility for Multiplication A D

The product \(A D\) is possible if the number of columns in \(A\) equals the number of rows in \(D\). Since both are \(3 \times 3\) matrices, the matrix multiplication \(A D\) is also valid.

Multiply A and D

Each element of the product matrix \(A D\) results from row-wise multiplication of \(A\) by corresponding columns of \(D\). The product matrix is:\[A D = \begin{bmatrix} 1 & 2 & 3 \4 & 5 & 6 \7 & 8 & 9 \end{bmatrix} \begin{bmatrix} -1 & 0 & 0 \0 & 2 & 0 \0 & 0 & 3 \end{bmatrix} = \begin{bmatrix} -1 \times 1 + 0 \times 2 + 0 \times 3 & 2 \times 2 + 0 \times 3 & 3 \times 3 + 0 \times 2 \-4 & 10 & 18 \-7 & 16 & 27\end{bmatrix} = \begin{bmatrix} -1 & 4 & 9 \-4 & 10 & 18 \-7 & 16 & 27\end{bmatrix}.\]

Compare and Summarize Results

The products \(D A\) and \(A D\) are not equal, which demonstrates the property that matrix multiplication is not commutative. The computed matrices are:- \(D A\): \(\begin{bmatrix} -1 & -2 & -3 \ 8 & 10 & 12 \ 21 & 24 & 27 \end{bmatrix}\)- \(A D\): \(\begin{bmatrix} -1 & 4 & 9 \ -4 & 10 & 18 \ -7 & 16 & 27 \end{bmatrix}\).

Key concepts

Diagonal Matrix

A diagonal matrix is a special kind of square matrix. Its defining characteristic is that all its non-diagonal elements are zero, while the diagonal elements are non-zero real numbers. For example, in the given exercise, matrix \(D\) is a diagonal matrix:

• The main diagonal consists of \(-1, 2,\) and \(3\).
• The rest of the elements in the matrix are all zero.

Diagonal matrices are important in linear algebra because they simplify many matrix operations.
When you multiply a diagonal matrix by another matrix, you can think of the diagonal elements scaling each corresponding row of the other matrix in the product. Understanding these properties makes matrix calculations more straightforward and less error-prone.

Matrix Product

The matrix product refers to the result of multiplying two matrices together. The process involves the following steps:

• Ensure that the matrices are compatible for multiplication: the number of columns in the first matrix must equal the number of rows in the second.
• For each element in the resulting product matrix, perform a sum of products of the corresponding elements from the row of the first matrix and the column of the second matrix.

In the exercise, the product \(DA\) is calculated by scaling each row of matrix \(A\) by the corresponding diagonal element of \(D\).
Meanwhile, for \(AD\), each column of \(D\) scales the corresponding row of \(A\). This often leads to different results for \(DA\) and \(AD\), showcasing the importance of order in matrix multiplication.

Non-Commutative Property

Matrix multiplication is different from regular arithmetic multiplication. In arithmetic, the order does not matter, as \(a \times b = b \times a\). However, in matrix multiplication, this is not generally true and is referred to as the non-commutative property.

• For matrices \(A\) and \(B\), it is not often true that \(AB = BA\).
• In our example, \(DA eq AD\). This demonstrates the unique nature of matrices.

This property implies that careful consideration must be given to the order of operations when multiplying matrices in any practical application, from solving linear equations to performing transformations in graphics.

Matrix Compatibility

Before you multiply two matrices, you must ensure they are compatible. A common error in matrix operations is overlooking this compatibility check. Matrix \(A\) can only be multiplied by matrix \(B\) if the number of columns in \(A\) is equal to the number of rows in \(B\).

• In the exercise, both matrices \(D\) and \(A\) are \(3 \times 3\), making both \(DA\) and \(AD\) valid operations.
• This step is crucial in preventing undefined operations.

Matrix compatibility ensures smooth execution in mathematical calculations, confirming that multiplication will result in a valid product matrix, which is vital for any further analysis or application.