Problem 29 - Algebra for College Students

Question

For Problems \(27-30\), use the following matrices. \(\begin{aligned} &A=\left[\begin{array}{rr} -2 & 3 \\ 5 & 4 \end{array}\right] \quad B=\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]\\\ &C=\left[\begin{array}{ll} 1 & 0 \\ 1 & 0 \end{array}\right] \quad D=\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right]\\\ &I=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \end{aligned}\) $$ \text { Compute } A D \text { and } D A \text {. } $$

Step-by-Step Solution

Answer

AD = \begin{bmatrix} 1 & 1 \\ 9 & 9 \end{bmatrix}, DA = \begin{bmatrix} 3 & 7 \\ 3 & 7 \end{bmatrix}.

Step 1: Understand the Context

The problem is asking us to multiply two matrices, \(A\) and \(D\), in two different orders: \(AD\) and \(DA\). Since matrix multiplication is not commutative, the results can differ. Matrix \(A\) is a 2x2 matrix and matrix \(D\) is also a 2x2 matrix.

Step 2: Write Down the Matrices

Recall the matrices\(A\) and \(D\):\[A = \begin{bmatrix} -2 & 3 \ 5 & 4 \end{bmatrix}, \quad D = \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} \]

Step 3: Calculate AD

To multiply \(AD\), use the formula for matrix multiplication: - Multiply each element of the rows of \(A\) by the corresponding elements of the columns of \(D\).- Calculating \(AD\):\[AD = \begin{bmatrix} (-2)(1) + (3)(1) & (-2)(1) + (3)(1) \ (5)(1) + (4)(1) & (5)(1) + (4)(1) \end{bmatrix} = \begin{bmatrix} 1 & 1 \ 9 & 9 \end{bmatrix}\]

Step 4: Calculate DA

Next, calculate \(DA\) by reversing the order of multiplication:- Multiply each element of the rows of \(D\) by the corresponding elements of the columns of \(A\).- Calculating \(DA\):\[DA = \begin{bmatrix} (1)(-2) + (1)(5) & (1)(3) + (1)(4) \ (1)(-2) + (1)(5) & (1)(3) + (1)(4) \end{bmatrix} = \begin{bmatrix} 3 & 7 \ 3 & 7 \end{bmatrix}\]

Step 5: Compare the Results

Notice that the matrices \(AD\) and \(DA\) are not the same:\[ AD = \begin{bmatrix} 1 & 1 \ 9 & 9 \end{bmatrix} \quad , \quad DA = \begin{bmatrix} 3 & 7 \ 3 & 7 \end{bmatrix} \]

Step 6: Conclusion

Matrix multiplication is not commutative, which means \(AB eq BA\) in general. We can see through this exercise that \(AD eq DA\).

Key Concepts

2x2 Matrices

Matrices are a fundamental element in linear algebra, serving as a versatile tool used across various mathematical and real-world applications. A 2x2 matrix is among the simplest forms of matrices, consisting of two rows and two columns. For example, matrix \(A\) in the problem is a 2x2 matrix:\[A = \begin{bmatrix} -2 & 3 \ 5 & 4 \end{bmatrix}\]Understanding the structure of a 2x2 matrix is crucial. Each element in a matrix is typically referred to by its row and column placement, often represented as \(a_{ij}\) where \(i\) is the row and \(j\) is the column number. These matrices are particularly important because they provide a manageable way to introduce matrix operations before moving onto larger matrices. In linear algebra, 2x2 matrices are often used for transformations in the plane, such as rotations and reflections.

Non-Commutative Property

One key aspect of matrix multiplication is its non-commutative property. This means that changing the order of the matrices being multiplied will lead to different results or sometimes even an impossibility. In simpler terms, for matrices \(A\) and \(B\), normally, \(AB eq BA\).In our given exercise, by calculating \(AD\) and \(DA\), we observe not only different numerical results but also the essential fact that the structure and dimensions in which multiplication happens change the entire product's outcome.This property holds fundamental importance since it underpins the careful ordering needed in many matrix multiplications, particularly those in computational and applied mathematics contexts. When applying matrices in processes like transformations or data representation, this property must be constantly considered.

Matrix Operations

Matrix operations, particularly multiplication, are foundational processes in linear algebra. When multiplying two matrices, you generally multiply rows by columns, totaling the products, and placing each result in its corresponding position in the resultant matrix.In the exercise, multiplying \(A\) and \(D\) is done as follows:- For \(AD\), elements are computed by multiplying rows of \(A\) by columns of \(D\), leading to: \[ AD = \begin{bmatrix} 1 & 1 \ 9 & 9 \end{bmatrix} \]- For \(DA\), the order is switched, with rows of \(D\) being multiplied by columns of \(A\): \[ DA = \begin{bmatrix} 3 & 7 \ 3 & 7 \end{bmatrix} \]Such operations highlight the need for precision in both calculations and comprehension of the multiplication structure, which includes dimensions like m×n by n×p multiplying to yield m×p matrices.

Matrices in Linear Algebra

Matrices are central to the principles of linear algebra. They are used for representing and solving systems of linear equations, one of the primary applications of linear algebra. They facilitate operations such as addition, subtraction, and particularly multiplication, which is essential for transformations, solving linear systems, and more. The significance of matrices in linear algebra extends to numerous multidisciplinary fields such as computer graphics, physics simulations, and data science. By mastering the manipulation and understanding of matrices, you gain powerful problem-solving tools. These fundamental components not only ease complex calculations but also enhance conceptual clarity, enabling learners to visually and numerically grasp multidimensional transformations and relationships.